Descriptive Geometry for Students of Engineering

D E SCRI PTI V E GE O ME TRY

FO R

B Y

M YE E A B E R . B E . B A AM R O S . . M J S OS . M , , , A ssistant P rof ess or of Mechani cal E ngi n eeri ng i n Charge of the Ill echa ni cal L abor a to r i es i n the Un i versi ty of IlI whigan; f ormerly I n str u ctor i n D escri pti ve G eo m etry i n Harvard Uni versi ty : E ngi n eer wi th th e Gen er al E lectri c Compa ny

d W ti h K r a nd Co m an a n wi th es ng o us e, Ch u rch , e r p y Member of the A mer ica n Soci ety of M echani ca l E ngi n eers ; Jl I i tgli ed d es Ver ei n es deu tscher I ngen i eure; M ember of Fran kli n I nsti tu te; B oston A ss oci ati o n f or th e A d vanc e men t of Sci ence; A mer i can Soci ety of Ci vi l E n gi n eers ; So ci ety f or th e n etc P romoti on of E n gi neeri ng E du cati o , .

T H I R D E D I T I O N

TH I RD THOU S A ND

NEW YO RK J OHN WILEY SONS

ND O N : C M I I E LO HAP AN HALL , L M T D 1909 i h t 1904 and 1905 copyr g ,

B Y R J AMES AMBROSE MOYE PREFA CE .

THI S book is the result Of teachi ng descri ptive geometry to

n to n b S students o f engineeri g . My aim is prese t the su ject O as to make it most easily applicable to the requirements of recent

n n i i n r n . n b engi ee i g practice The methods Of prese tatio th s ook ,

no t di n . n n therefore , are tra tio al Experie ce has show that most studen ts i n ou r best techni cal schools have difﬁculty in applying their knowledge o f this subject to subsequen t work in structural

n n hi n b n n and machi e desig . Two t gs have ee attempted i this book to overcome thi s failure Of o ur studen ts : (1) The notation

as in ni n F r is essen tially the same that used mecha cal drawi g . o

n n and ri a lo g time , practical drafti g desc ptive geometry have had 2 to o little in common . ( ) The exercises have been carefully

to n n i nk n fo r s and graded e courage a stude t to do th i g him elf ; ,

his n n n n to stimulate i terest , ma y co crete exercises , showi g usually

b n n ns s . n practical applicatio , have ee i erted Such exercises , I thi k ,

be n b nni n so should i troduced from the egi g , that the student may

n o f b n see the practical applicatio his pro lems as he goes alo g . The data for the exercises are stated by the system Of coor

n n R dinates used i a alytic geometry . easons for choosing this

_ F r b b . o i nn n hi b system are O vious a class eg i g t s su ject , there is a great advantage in stating the exercises wi th absolute deﬁni te

ni b is no t i n n . n n in ess If a deﬁ te pro lem g ve , ma y stude ts , order i n to Show a sat sfactory solutio , will waste much time selecting data ; and others will present drawings that fo r their complication b are mostly unintelligi le .

“ 2 3638 4 n use n r n Illustratio s are of more tha much wordy desc iptio . Fo r this reason an unusually large number of perspective and

n n n l n in orthographic drawi gs have bee i serted . The i lustratio s

n b perspective are very helpful . Whe ever it is possi le , however , studen ts should be en couraged to make models o f bardbo ard and “ n W n pe cils that they may build hat they are drawi g .

- n L n This book is no t intended for self ins tructio . ike la guages ,

and this subject can be learn ed successfully only from a teacher , " n n b and s n o t alo e from ooks lecture . The stude t must take the

n has b n o n time to work out ma y exercises . Space ee left the right

n n and n ha d pages for lecture otes Sketches . The stude t may well

n n o n put the solutio s for ma y of the exercises these pages .

A f n to n b s 6 nd O 7 a 8 . good deal space is take explai Pro lem , , These are considered fundamental ; and the teacher Shoul d be sure

" b n they are mastered efore the stude t goes further . With these

b i n n be no w pro lems well mi d , there should difﬁculty ith those

‘ b n b n n that follow . It has ee my O ject to make the expla atio s Of the problems throughout the book consisten tly briefer as the

b - s su ject matter i developed .

n b n H and I am u der great O ligatio to Professor Ira N . ollis

L w n n fo r n and n n Professor e is J . Joh so much assista ce e courageme t i n ri n b o w e n n prepa g this ook . I Special ack owledgme t , however ,

H n b n i n i . aco i b to Professor e ry S J y , who led teach g th s su ject

l n He with its practical app icatio s . has carefully read much Of

b and n n this ook , I have received ma y suggestio s from him .

Fo r n i n n b assista ce ma y ways I wish to thank my rother, Mr .

. n f i a h B L o f . . O i . J Clare ce Moyer, M E , Ph l delp a , Mr C . . ewis

n nn nd B n a . O f b Ci ci ati , Mr rya t White Cam ridge . A R Y . J . . MO E

C B RI G E D cem er 15 e 03. AM D , b , FA T T HE S ND D I P RE CE O ECO E TI ON.

T HE gratifyi ng results with the ﬁrst edition showed that the

O f b b n n methods this ook were appreciated eyo d my expectatio s ,

n I n the second edition I have added a umber O f new exercises .

Many O f these appear throughout the text .

n n O f A I n r n . i . . n prepa g the seco d editio the help Mr E Norto ,

F r b h B b n n b . o n P . and . , has ee i valua le to me valua le suggestio s

n b n B n criticisms I am much i de ted to Comma der arto , U . S . Naval

n n A s. . . K n A . n cademy ; Prof dams , Mas I st of Tech ology ; Prof e edy ,

n n n . H n . . R n B arvard U iv . Prof Ogde , Cor ell U iv Prof a dall , rown

f nn n n n n . O . . . n n U iv Prof Spa gler , U iv Pe sylva ia ; Prof Tilde , U iv .

n Y n and . . . V Of Michiga ; Prof Tracy , ale U iv ; Mr W . . Moses of

n n the Ge eral Electric Compa y .

l The American B ridge Company and the B oston B ridge “o rk s have kindly supplied drawings from which the data for some O f the

n n exercises have bee take .

’ in Ferris s b I am much gratiﬁed that Prof . ook o n descriptive

has b n b an ff geometry which just ee pu lished , e ort is show n to

. i n e c n . in meet , a d gree , pra tical requireme ts S ce the ﬁrst edi tion

b n Of this ook appeared , I have received ma y letters regarding the . relative importan ce to be given this subject from a practical View

n n n n n n n poi t i a course i e gi eeri g . These i quiri es in terest me much , and in i n I n i reply g have gladly give the results Of my exper en ce .

. A . Y J MO ER.

B E n r 1 9 C RI G a u a 05 . AM D , J y , FA T THE H RD D PRE CE O T I E ITION.

US RI n b n IND T AL educatio is ecomi g , every day , more important n n i i n all systems Of teaching . The te de cy n education is toward n the economic applications . The adva tages of teaching with the help of practical problems and exercises is more appreciated than

n n . new ever , with correspo di gly more satisfactory results These n in b requireme ts are measured , a degree , y the success Of this book . I n n n b r this editio some cha ges , mostly suggested y teache s , b n i n and an n h as b n to have ee made the text , i dex ee added make

r n F r the boo k more convenient fo refere ce . o very valuable criti i . L l O f B n r cisms n b . nse Ge I am especially i de ted to Prof Dr erli ,

. M n and . acob of n ma y , Prof J y Of Ithaca uch Of the work revisio

h as n to . A . . n b falle my colleague Mr E Norto of Cam ridge , whose services I cannot to o highly appreciate .

A . . J . MOYER

Y Dec m r 1 N e e 906 . L N , b , CONTENTS

I NT RO D O U CTI N 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

CHAPTE R I

E LE ME NTARY PRI NCI P LE S AND NOTATI ON

The Pl anes o f Proj ectio n

ro ction s o f o ints and ines P j e P L .

CHAPTE R II

R B LE E L I G T H E I I E E P O MS R AT N TO PO NT , L N , AND PLAN

Probl ems an d Practical Exampl es

CHAPTE R III

PRO B LE MS RE LATI NG T o PLANE S TANG E NT T o SOLI DS

rf Classiﬁcatio n o f S u aces . Convol ute Surfaces

CHAPTE R IV

INTE RSE CTI ONS AND D E V E LO P ME NTS O F SOLI DS

I ntersectio ns o f Surfaces w ith Pl anes

m n f u rfac s D evel o p e t o S e . Intersectio ns o f Surf aces

CHAPTE R V

E U R C I C L E xE RCI SE s MI SCE LLAN O S P A T A . CHAPTE R V I

SHAD E S AND SHADOW S

CHAPTE R VII

VVA RP E D SU RFA CE S

CHAPTE R V III

PE RS P E CT I VE

The Prin ciples o f Perspective D raw in g

rs cti D sto rtio n Pe pe ve i . Perspe ctive Sketches fro m Wo rking D raw ings

r e Practical Exe cis s. D E SC RI P T I V E G E O ME TRY

INTROD UCTION

‘ scr e G th e metho ds n I . D e iptiv eometry treats of of maki g draw ings to represen t objects w ith mathematical accuracy . There are

n n B o ne two common methods for such represe tatio . y method ,

n e all ed perspective drawi g , the chief purpose is to produce a pictur which will be plain to a person unfamiliar wi th the methods used B for technical drawings . y the other method , however , the chief aim is to Show an Object with the true dimensions that are n eeded

n n in the construction O f buildings and machi es . The drawi gs are then made by a method which does no t give a pictorial effect ; f b m u t o n the O n O , b b , ther ha d , shows views the O ject , fro which y n n can be very Simple processes , true dime sio s Of all parts quickly h r a c e n . Obtain ed . This latter method is called o thogr p i proj ctio

I t is ~the method with w hich the studen t must become most

‘ and c n en hi m familiar , with which this treatise must most o c r

s n be . for some time . Per pective drawi g will discussed later The method Of orthographic projection represen ts the outlines Of the Object as they might be traced o n transparent plan es placed around

1 n an d 2 a n i n F . the Object as show igs (fro tispiece) , where three views o f a hexagon al pyramid are Shown pictorially o n horizon tal

D w n n n i b and vertical planes . ra i gs represe ti g these v ews y the orthographic method are made in the same way as i n mechani cal

b is n drawing . The O ject thus represe ted as though the eye were

n ni n l n b inﬁnitely dista t ; that is , the va shi g Of the i es Of the O ject

n in the distance is not represe ted . CHAPTER I

E LE ME NTARY PRI NCIPLE S AND NOTATION

2 ri n and n n . The ho zo tal two vertical pla es upo which the

n i n Fi 2 a three vrew s Of the pyramid are show g. are called the

s f r e n n plan e o p oj ctio . These pla es are always taken at right angles to each other and are designated according to their position

h n n c nd d e a a a a . n as orizo t l , fro t verti l , si e v rtic l The li es Of i nter section o i the horizon tal with the front vertical and side vertical

n X and Y n pla es are called respectively the axes . The i tersection

n and n Z Of the fro t vertical side vertical pla es is called the axis .

n n i n and n These axes are show plai ly the ﬁgure , the poi t where

n n and is they i tersect is called the o rigi , usually marked O .

r I n Fi 2 a n n n in 3 . g. the pla es Of projectio are show a pictorial

n n is drawi g , where they are placed arou d a pyramid which the

t n n n Object o be represe ted . The pla es are arra ged as we must

n h n and imagi e them placed to S ow the top , fro t , side views Of the

n the n n n i n e pyramid , accordi g to co ve tio al methods used prac

I n mi tical drafting . this ﬁgure the views Of the pyra d Shown o n n n n b O f the pla es Of projectio , are its outli es made y rays light reﬂected from points o n the pyrami d perpendicular to a plane

n n n Of projectio . The poi ts where the rays pierce these pla es are called the proj ection s of poin ts o n the surface Of the pyra

i n n O f mid . Thus , the ﬁgure , two cor ers the pyramid are marked a nd b F n n n n n a . rom these poi ts dotted li es are draw represe ti g rays of light reﬂected from them perpendic u lar to th e plan es

f n n n n o proj ectio . The i tersectio s Of these dotted li es from ‘ h n a at and as a and b with the pla es are marked respectively , ,

' 8 h f h se th e ro ec and b b and b . t e , , Of , ﬁrst three are called the p j

h e n a and ro ec ns o f th e tion s o f t poi t ; the last three , the p j tio

n c n o f n n i n poi t b. The proje tio s other poi ts are fou d the same wa SO n O f y , that the complete projectio s the outlines o f the pyra

can be o n n n mid made each Of the pla es Of projectio . The co m pl ete projections o f the outlines Of an object are called its h o ri z ontal n erti cal and s de erti cal ec n s , fro t v , i v proj tio . Fo r objects

n n nl ns o f n with a gular outli es , o y the projectio the cor ers are

s n . b n o f u ually eeded The views Of the o ject are the , course ,

n b n n n n draw y joi i g the projectio s Of the cor ers . The dotted

n in n n n o n li es the ﬁgure , represe ti g the rays from the poi ts the n l n m ns ec es. pyra id to the projectio , are called proj ti g i

i 2a n O f n I n F . n g we see the pla es projectio , the hexago al pyra mid and n o f n o n , the projectio s the outli es Of the pyramid the

in All n n n as . o e pla es , a picture is show as complete view , such as we see when several sides O f an Object we are looking at are

n n n . SO see from a si gle viewpoi t If , however , the eye is moved

' n o ne n o f diﬁ eren that we see o ly projectio at a time , a set three t

be b n o rth a h c ec n s B views would O tai ed , called ogr p i proj tio . y the

o f n u be n methods orthographic projectio , these views wo ld arra ged

2 I n n n n i b. n as i F g. this latter ﬁgure , a horizo tal li e is ﬁrst draw

n n O f to represe t the X axis , usually ear the middle the space to

n n n n be taken for the drawi g . The at right a gles to this li e the

n n n Y and Z axes are draw as show . To represe t by thi s method

i n i 2 a z n F . n n the views show pictorially g , the hori o tal projectio is drawn behi nd the X axis and o n the left- hand Side Of the vertical

i n n n b Y ax s . The fro t vertical projectio is draw elow the X axis

n n and vertically below the horizo tal projectio . The side ver tical projection is also drawn below the X axis but o n the other

As n z n side Of the Z axis . the views are thus show the hori o tal projection shows the outlines O f the pyramid when the eye is im

n n n and mediately over the horizo tal pla e . The fro t side vertical projections represen t the outlines as they appear when the eye

n n n is moved in front Of the fro t a d side vertical pla es . This is a

n n n n O f an b and very conve ie t way to arra ge the projectio s O ject , it has the advan tage that it is more easily understood by mechanics

n b n n n o f Of ordi ary a ility , tha some other arra geme ts the same projections that will be discussed later .

in i 2 a i 3a n F . I n F g. the same pyramid that is Show g is rep resen ted in a differen t position with respect to the plan es of pro

‘ ec io n I n Fi 2 a below he h ri z n l j t . g. the pyramid is placed t o o ta

n n n n n in Fi a v a d b . 3 abo e pla e ehi d the fro t pla e ; g , it is placed

n n an n n n Fo r n the horizo tal pla e d behi d the fro t pla e . this arra ge

n n b n n in me t , the drawi g made y orthographic projectio is show i 3b F . . g This method , however , is scarcely ever used , as it crowds

n n the projectio s too much for ordi ary purposes . Observe that in the orthographic projection the front vertical projection is above ’ x and the ho ri z o ntal n b n b the X a is , projectio is ehi d as efore . It

ns n n b thus happe that for this method Of arra geme t Of views , oth the horiz ontal an d the vertical projections are shown i n ortho

n O n O f I n Fi 4 . . a graphic projectio the same side the X axis g ,

b n n n H however , a little etter arra geme t of views is Show . ere the pyramid is above the horiz ontal plane and in front o f the fron t

n I n n n pla e . orthographic projectio the same views are show i n

i 4b n n o f n n F g. . This arra geme t the projectio s was o ce much used n n t i i n . o n Fi b 3 . practical drafti g It does crowd the views as g.

i 5a below n n n in I n F g. the pyramid is the horizo tal pla e a d front f n n n O f O the fro t pla e . Orthographic projectio s these Views are

n in Fi 5b n n O f n i show g. . This last arra geme t the proj ectio s s also

n and h as no n n i n likely to produce crowdi g , Sig iﬁca ce practical

n drafti g . It should be mentioned here that the arrangemen t o f Views Shown Fi 2b by orthographic projection i n g. is adopted almost exclusively nt fo r engineering drawings in the United States . This arrangeme

n n n B y is almost universally applied i n moder machi e drawi g . this

i n n n u method , the views are placed the most atural positio s to S ggest a mental picture Of the Object represented by the drawings ; the to p

n and n in projection is at the top Of the drawi g , the fro t View just

- n fron t of the top view . The right ha d side view is at the right

n han d side of the Sheet . The views are thus arra ged where they

n can best suggest a mental picture to a workma .

n n Fi 2b b 4 . The actual method of maki g a drawi g like g. y

n c n be n no w be . a orthographic projectio , will take up It most easi

i n ly explained by Showing how the prin cipal points n the drawi g

A n of were laid out. cor er the pyramid marked a has been referred h to b and its n a af and as n o u t efore ; projectio s , , were poi ted .

s in h in i 2 b They are al o marked the orthograp ic projections F g. .

w n b n s from X Y These projectio s are located y their dista ce the , , h Z n n a and . z n o axes Thus the dista ce from the hori o tal projecti , ,

to l Y s o z n o n the vertica axi , measured h ri o tally , sh ws the dista ce , in n a to n n Space , from the poi t the Side pla e . The dista ce from h a to u i n the X axis , meas red vert cally , shows the dista ce Of the

n a b n n t n n t t n poi t ehi d the fro vertical pla e . The fro t ver ical projec io , af is b an d n , elow the X axis , the dista ce , measured vertically , from

is n to n n a th projectio the axis , shows the dista ce that the poi t , ,

s n n . n a is below the horizo tal pla e The Side vertical projectio , , is

n b in off i n z n to fou d y lay g a d sta ce , measured hori o tally , equal

n n a b in n n the dista ce that the poi t is eh d the fro t vertical pla e .

tw o n af and a8 n e Of course , the projectio s are at equal dista c s below the X axis.

o f descr t e o r racti cal e me 5 . The methods ip iv p g o try are useful to engineers and architects in many structural and mechanical

n I n ni n and n n b i n and n Operatio s . desig g co structi g u ldi gs machi es , it is Often necessary to ﬁnd the true size and Shape Of parts Shown

n h n n n n n no t n o n drawi gs . W e the ecessary dime sio li es are show ,

n n fo resho rten ed indi rect and whe some Of the actual le gths are , _

n geometrical methods Of measureme t must be used . The study

fo r n n Of this subject is useful , however, more tha its i dustrial utility . The student becomes accustomed to considering very complicated

' n ns and to accuratel t he n geometri cal combi atio , follow y correspo d “ n b n ence between the drawings a d the O jects represe ted . De “ - n n n n has n o ne scriptive Geometry , a well k ow e gi eer said , trai s ” n n in to see around cor ers . I deed it exercises the most precise

to i n n to s manner the power v sualize , which is represe ti g ourselve , b b . clearly and easily , ideal O jects as if they were really efore us The importance o f this subject in training students fo r work in b l n and n can be designing b ridges , ui di gs , machi ery hardly over

e if th e stu d o f thi s sub ect d es n o t estimated . Neverthel ss , y j o

hi nk th e al i s m ssed. teach th e student to t , go i

The methods Of d escri p tive geometry are absolutely general in t a l n t t in w k heir pp icatio so ha , the or that follows , if the solution o f a p roblem is given for any o ne o f the arrangements of views t b n n b hat have ee explai ed , it is applica le as well for all the others .

Th e an es o f ec n are nde n e in e en pl proj tio , also , i ﬁ it xt t ; that is ,

t n t in n P c n s f they ex e d withou limit every directio . roje tio o li nes m a be r d c d as far as t are n eded en th y p o u e hey e , wh e solution o f

r a p oblem requires it .

6 P es n n t n . lan other tha the pla es Of projec io must be O ften

n An n can be n n represe ted . y pla e that Show withi the limi ts o f

n i n o ne o r o f n the drawi g , w ll i tersect more the pla es Of projection

f n in s l n . n o n ces f traight i es These li es i tersectio , called the tra o

n n n i n the pla e , are made use Of to represe t pla es orthographic pro

An b n n r n jectio n drawings . O lique pla e i te secti g the three planes

i n i 6a n i n i n F . n f o f project o s show g . The i tersectio o the plan e with the horizon tal plane Of projectio n is called its h orizo n tal

n n n and trace . Its i tersectio s with the fro t side vertical planes

its r n al and s de a a e are called respectively f o t vertic i vertic l tr c s .

n is n i n Fi 6b b n The same pla e Show g. y its traces , as it is represe ted

Fo r b t O f n . b y the me hod orthographic projectio revity , however ,

h r z nd de be n n a s . these traces will called simply o i o tal , fro t , si trace The Simpler name for the last two traces can lead to no confusio n

n r y F r and short names are preferable to abbreviatio s o s mbols . o

the s n n n l be t same rea o , the pla es Of projectio wil called , hereaf er ,

h r n n and s de an e and w e simply o izo tal , fro t , i pl s ; shall use the

n n n fo r n o f n and n correspo di g simpler ames the projectio s poi ts li es .

NOTA TI ON .

ect n s a nt in n b 7 . The proj io Of poi Space are desig ated y a small

w t h o r s b n w n letter i h , f, placed a ove it to i dicate hich projectio is h z n a nt and s de ect n o f a oin t meant ; thus the ori o t l , fro , i proj io s p h a af and as I n the n a are marked respectively , , . drawi g , these

projections are located by the di stances o f the point a from the

o f n to the X Y and Z a es planes projectio , measured parallel , , x ;

in o f i n n xe and the geometry space , the d sta ces alo g these a s are

c rdi nateS as and v b oO z . represented respecti ely y the , y, Di stances al ong th e X axi s ( represented by th e a: co ordinates) h l t r ri h fr th e s de l ne L n are measured to t e ef o g t om i p a . o g usage h as establish ed th at th ese distan ces sh al l be con sidered negative wh en measured to th e l eft o f th e sid e pl an e an d posi

r t th e r ht tive wh en measu ed o ig . D istan ces al on g th e Y axis ( represented by th e y co ordinates)

r ur b hi n d r in r nt o f th e fr nt an e Th a e meas ed e o f o o pl . ese are n egative wh en behi n d and positive wh en in front of the front

l an p e . Di stan ces along th e Z axi s ( represented by th e z coordi nates) are measured below o r above th e hori z ontal pl an e ; n egative

wh en bel w s e w h en ab e . o , po itiv ov

n is n as: a — 2 — 3 n If a poi t represe ted , , we mea that the

n a has fo r c rdi nates a: = — 3 and z = — 4 — poi t its y , ; that

n a 2 ni n to o f n 3 n the poi t is u ts Of le gth the left the side pla e , u its b n n n and 4 n s b z n n ehi d the fro t pla e , u it elow the hori o tal pla e .

8 r ect n s of a line n te . The p oj io are de o d by the projections O f two o r more points in the line ; thus a line containing the two points a and b is called the line a b and is represented by the co br tw in dinates o f these o po ts . The poin t where a line intersects the horizontal plane Of pro jecti on is marked by the letters h i ; and the intersections o f a

n n and lan s b i and s TO li e with the fro t side p e y i i . locate these points more plainl y draw a small circle around their projections

in i and 9 on es 2 1 and as n F . 8 2 . show gs , pag 9

traces O f an es s n e in wa 9 . The pl are repre e t d a y suggested by i f r n co Ord nates o s . A n n in Fi the method Of poi t pla e is show gs .

b The n and n in 6a and 6 . horizo tal fro t traces tersect on the X axis to the left o f the origin and make with this axis the angles

B b n m e a and . s n ark d ﬁ y ym ols such a pla e, P, is represe ted as ° a his n n fo r n is b b s , T otatio pla es remem ered e t by observi ng that the ﬁrst number (the x coOrdinate) represents the in tersection o f the traces o n the X axis ; and that the second and thi rd refer to the number o f degrees the horizontal and front

n in n traces make with the X axis . These a gles are measured a ti

o ec n i n n I n Fi 6b n a cl ckwise dir tio as trigo ometry . g. the a gles

° and ﬂ are respectively about 20 and I n such a n otatio n

n o t n n for the traces , it is ecessary to desig ate the side trace , as an mi n n and n n n y two traces deter e the pla e, data co cer i g a third

I n . n n is usually superfluous the drawi gs , the traces of a pla e are in nami n n H marked with the same letter used g the pla e , with ,

r w n i F r in i F o t b t. o F b . 6 , S ri te efore example , g , the hori z ontal n and n , fro t , side traces of the pla e , P , are marked

H F and respectively P , P , S P . I n all drawin gs the horizontal and fron t traces must in tersect o n i and n f r n n o n the X ax s , testi g o this i tersectio is a check the

f n accuracy O the constructio s .

1 0 . L n and O f n n re re i es traces pla es , give or required , are p

n in n b n n b b se ted orthographic projectio s y full li es whe visi le , y

n n n n b dotted lines when i visible . Projecti g li es are i dicated y

O f n n short dashes . Traces auxiliary pla es are usually represe ted

o by ne long and two Short dashes .

CHAPTER II

O E MS E G TO THE O N NE AND E PR BL R LATIN P I T, LI , PLAN

I I n n n n in e . The relatio betwee the actual positio of a li e spac and n o n a d n n n t t its projectio s b rawi g , is easily co ceived for a li e ha f n if w to n O . is parallel a pla e projectio D ﬁculty is , ho ever , usually experienced in con ceiving this relation for a line that is Oblique to

‘ n A n n is all the planes Of projectio . pictorial drawi g of such a li e

i a ns n n b n n n in F 7 . Show g. The projectio Of the li e are draw y joi i g

n f n w n in n Fo r n n the projectio s O a y t o poi ts the li e . the li e show in n n in n the ﬁgure , the poi ts most easily determi ed the drawi g are

n nd n n n its intersections with the horizo tal a fro t pla es . These i ter h h nd bi a n t n a . sec io s are marked a The former , , has its horizo tal

t n n n w n and its n n projec io coi cide t ith the poi t itself, fro t projectio f in at a . tt n t n is , of cours e , the X axis Similarly the la er i tersec io , h bf n and n n b in , has its horizo tal fro t projectio s at ( the X axis)

n bf nt and n ns n a d at . The horizo al fro t projectio Of the li e are ” " n n b and af w bf then drawn by joi i g a with ith .

1 2 B n n n o f n w . y reversi g the process the i tersectio s the li e ith the plan es o f projection can be fOu nd when the projections O f the

in i 7b n t n n a b line are given ; thus F g. the i tersec io Of the li e with

n n n a n b n th e horizo tal pla e is the poi t , , Show y its horizo tal pro h ectio n a b n n n n n j , , ( ehi d the i tersectio of the fro t projectio Of the

n n af in i and . lin e with the X ax s) its fro t projectio , , ( the X axis)

n n n n n is n The i tersectio Of the same li e with the fro t pla e the poi t , " b ns I) and bf b n . , show Similarly y its projectio Wh en on ly tw o pl anes o f proj e ction are m ention ed ( as i s

st r b ems i s assu th e s de an e i s n o t the case for mo p o l ) , it med i pl

needed .

1 1 To draw the ro ections o a li ne havi n i ven 3 . PROBLEM . p j f g g

s the i ntersecti ons of the li ne wi th the hori zontal and front plane .

h d h n n n Met o . Draw t e horizo tal projectio Of the li e by joining n n the horizo tal projections Of the two i tersections . Draw the front projection Of the line by joini ng the front projections o f the tw n n o i tersectio s .

E XERCISE S

F r n n see Arts — —Th e c din ates f nt r o otatio . 7 9 . oor o poi s a e

x r s in in h s e p es ed c e .

1 a A n n e n n n . ( ) li e i t rsects the horizo tal pla e at the poi t a

— 3 — 2 0 and n n n b — 1 0 , , ) the fro t pla e at the poi t , ,

n n Draw the projectio s Of the li e .

b A n h n e — 2 —2 and ( ) li e passes t rough the poi t , ,

n n and n n o 0 0 i tersects the horizo tal fro t pla es at ( , ,

L ns n ocate the projectio Of the li e .

I 2 ; v the o ect ons o a l ne o he o . R M . Gi en r i i t nd t i nt 4 P OBLE p j f , ﬁ p where the li ne i ntersects a the hor z ontal lane b the ront lane i . ( ) p , ( ) f p

e d For a n n n o f M tho . ( ) produce if ecessary the fro t projectio

n to ront n o f the li e the X axis . This is the f projectio the required

“

n i n n TO ﬁnd hori zontal n . poi t Of tersectio . the projectio , draw a

r n i i nt i n t n pe pe d cular to the X ax s through the fro project o jus fou d , to meet the hori zontal projection Of the given line; Fo r (b) pro

n n f n ‘ duce the horizo tal projectio o the li e to the X axis . This is

z ntal n f i n o f n n The the hOri o projectio o the requ red poi t i tersectio .

n f n n a n front projectio o the poi t is the e sily fou d .

EXERCISE S

2 n n n ornts c 3 —1 — 2 . Give a li e passi g through the p , , )

and d 1 — 2 F n n n , , i d the poi ts where this li e

n 1 n n 2 n ne i tersects ( ) the horizo tal pla e , ( ) the fro t pla ,

(3) the side plan e .

a ns tw o n o ne to the 3 . ( ) Draw the projectio Of li es , parallel

n n and b to n n horizo tal pla e O lique the fro t pla e , the other parallel to the fron t plane and Oblique to the horizontal

(b) Find the intersections o f each line wi th the horizontal

and n n f n fro t pla es o projectio .

n n n in 2 O . n k 1 n b the li e give Ex , locate a poi t , , i ch elow

n n and n n l i n the horizo tal pla e a other poi t , , i i ches

b n n n w n ehi d the fro t pla e . Sho three projectio s of each

n poi t located .

G n n n e — — 4 5 . 3 1 2 and ive the li e through the poi ts , , ) f — — 1 2 L n o n n , , ocate its projectio the side pla e .

A an lirie in an 6. d n n . F ssume y space , draw its projectio s i d the new projections o f this lin e (a) when the X axis is perpendicular to its former position (coin ciding with the ° Y i b n X 3 vertical ax s) , ( ) whe the axis is revolved 0

n - r n (a ti clockwise) f om its ﬁrst positio .

est o n is n n S ugg i . The exercise easily solved by ﬁ di g h i

and i o f n Wh n n n f the li e . e the pla es of projectio are

n n revolved , the positio s of these poi ts with respect to the

ndt be n axes will cha ged .

1 A n i n in n n n r n n 5 . li e ly g a give pla e i te sects the horizo tal pla e

f n n i n z n in n o projectio at a poi t the hori o tal trace ; that is , the li e showing the intersection of the given plane with the horizontal

n n n n n n in pla e . The same li e i tersects also the fro t pla e at a poi t

n o f n n s b the fro t trace the give pla e . This is illu trated y a pictorial

i n n . d i n in Fi . 8a n a b n n raw g g where a li e , , lyi g a pla e , P , is show The line is here produced upward u ntil it intersects the hori z ontal plane at h i and downward to intersect the front plane at f i . Fi The problem n o w takes this form (see g. 8b) z Given only

n n af bf n a b i n h n a the fro t projectio , , of the li e , determ e the orizo t l h h n h n a b i n n ec a b so t e . proj tio , , that li e shall lie the pla e P h h Instead of ﬁnding at once the points a b it is ﬁrst necessary

b h e n s th e n h i and i to locate ot proj ctio of poi ts , i , where the given line a b intersects the horiz ontal and front planes respect

n z n n the n hi and i i vely . The the hori o tal projectio s of poi ts i

h n oined th e n n n . provide , w e j , , required horizo tal projectio of the li e Obvi o u sly th e h o ri z o n tal p ro j ectio n o f th e p o i nt h i i s i n th e h o ri z o n tal tra e o f th e l an e P an d the f ron t ro ecti o n o f th e sam e o int i s i n th e c p , p j p

X a i s imilarl the fro nt and h o ri z o n tal ro ecti o n s o f i are res ecti el x . S y p j f p v y

n ra nd i n X a s I n s i n the fro t t ce o f P a th e xi . th e ﬁgure the pro j ecti o ns of

h i an d i i in th e . X axi s are n o t m ar k ed to avo id confu sing th e drawings w ith m s to o any l etter .

O n n n the n b n h i and i the horizo tal projectio of li e etwee f , h h n ns a and b b thus fou d , the required projectio are located y pro

' n s n X i f f j ecting li e perpe dicular to the ax s through a and b .

1 6 Gi ven one ro ecti on o a li ne or n . R 3 . a oi t i n a P OBLEM p j f , p ,

i ven lane to nd the other ro ecti on . g p , ﬁ p j n s n n f i n e a G o . n M thod. ( ) ive o e projectio a l e Determi e the po sitions o f the intersections o f the li ne in the given plane with

d n n an n the n an d i . horizo tal fro t pla es , draw the requ red projectio

n n n f n r e hod b G o e o . o M t . ( ) ive projectio a poi t Draw a p

in n n f n F n jectio n o f any l e through the give projectio o the poi t . i d

n o f i n as b b At n the other projectio this l e descri ed a ove . the i ter

n o f n b n n n sectio the projectio that has ee fou d , with a projecti g

n n n o f n the ro li e from the give projectio the poi t, is required p

i n ject o . E XE RCI SE S — n a 2 and b — 1 i s in 7 . The li e through , , a 0 n and plane P ( , Draw the fro t side pro

n jectio ns o f the li e .

n o ne n o f n b 8 . Give projectio a li e which is o lique to the X

in n to h . F n axis a pla e , Q , with traces parallel t is axis i d

another projection of the line .

n m in n n in . 7 9 . Locate a poi t , , the pla e , P , give Ex that

n n I n shall be inch behin d the fro t pla e . the same

n in n 1 » n b pla e , locate also a po t , , 1 i ches elow the hori

n z ontal pla e .

n o f nt 0 in n R Draw the three projectio s a poi , , a pla e , ,

f n which is perpendicul ar to tw o planes o projectio .

n o f in at in n S Draw the three projectio s a po t , a pla e , ,

n which is parallel to the horizon tal pla e .

The s10pe o f a roof o n a building is represented by its hori

s i n n z ontal and front trace , mak g a gles respectively of 0 ° in the 150 and 225 with the X axis . There is a hole roof for a chimn ey of which the horizontal projection is " f shown in the drawings as a regular hexagon (length o

n sides is i ch) . The horizontal projection o f the cen ter of the hole is located 1? in ches to the left o f the in ter

n o f and n b n sectio the traces i ch ehi d the X axis . Draw

the n n fro t projectio of the hole . (The required projec tion is found readily by drawing the projections of the

n diago als of the hexagon . ) State the usual notation fo r

n A r the pla e of the roof ( t. A 13 . of 1 n circular water pipe , which the diameter is i ch , passes through an inclined ﬂoor represented o n a draw ing by horizontal and fron t traces makin g angles respect ° ° i el 150 and 2 10 h v y of with the X axis . T e center lin e

o f n n n the pipe is parallel to the horizo tal pla e . The fro t

n and projectio of the pipe is therefore a circle , its center is shown 3 inches to the left o f the intersection of the

and ? n b traces 5 i ch elow the X axis . Complete the draw ing by ﬁnding the horizon tal projection of the hole

D n through which the pipe passes . ( raw a umber of

n n n diameters of the circle, produci g them whe ecessary, ﬁ n n and nd the horizo tal projectio s of these lines . The points in the outline of the horiz on tal projection of the hole will be vertically over the corresponding poin ts in

n the gi ven front projectio . )

z n n n 14 . The hori o tal trace of a pla e i tersects the X axis at — x = 1% and makes with it an angle of A point — — — 4 i D 1 1 n n . n nt 0 , 5 ) lies the pla e etermi e the fro

n D 0 an n trace o f the pla e . ( raw through y li e that will have its horiz on tal in tersection (h i ) in the horizontal

i n n n . trace . The i of this li e determi es the fro t trace )

To draw throu h a i ven oi nt a li ne arallel PRO B LE M 4 . 1 7 . g g p p

e to a gi ven plan . d o ne n an li ne and b Metho . Draw projectio of y , y the method

n e n o f the last problem , determi e its oth r projectio so that the

n n line will lie in th e given pla e . Through the two projectio s of

' ven oi nt n n the gi p , draw the projectio s of the required li e parallel to the corresponding projections of the line which has been drawn

n n t and n in in n . n the pla e The li e through the poi the li e the pla e,

n h b n an show t us y parallel projectio s , are parallel to each other ; d

h n n n b n to n n i n t e li e passi g through the poi t , ei g parallel a li e lyi g

n n . the pla e , is parallel to the pla e

P a ra ll el l i nes h a e aralle l ro ectio n s o n an lane v p p j y p .

E XERCISE S

— — r n a 3 1 n 15 . Th ough a poi t , , draw a li e parallel to

n R — 3 the pla e ,

n n n o f the n b — 16. Draw through the poi t of i tersectio li e 3,

— - 0 c l s n n , % , {) with the horizo tal pla e , n n T 0 a li e parallel to the pla e ( ,

F n n n n 8 i n 17 . i d the i tersectio with the side pla e , , of a li e with

n l both projectio s paralle to the X axis . — F n n n o f n d 3 e 18 . i d the i tersectio the li e ,

7 1 1 t n . i n , 3) wi h the side pla e Through th s poi t draw the side projectio n of a lin e which is to be parallel to the plane U o f which the horizo n tal and front traces are lines

o 1 n h parallel t the X axis . The former trace is 5 i c es

behind and the latter is 2 in ches below the X axis .

t n a b in F 8 R b . 8a an 1 . emem er tha the li e , , igs d 8b which are i n n o n n n . b r agai show the Opposite page , is the pla e P O se ve ,

n n h i and i o f i n also , that the i tersectio s , i , th s li e are respectively

n and n n i n the horizo tal fro t traces of the pla e . No w if the pro

ectio ns O f n n in n n n j a seco d li e the same pla e were give , its i ter

‘ ns h i and i u be n in the lane sectio , i , wo ld poi ts also the traces of p .

n n t n b an The suppose the co di io s of the pro lem are reversed , d the

o ecti ons o f n in n v n and c pr j two li es a pla e , P , are gi e we are to o n

' H an F traces d . n an struct the , P P The horizo tal d fron t inter

n h i and i o f n n Th sectio s , f , each li e are usually very easily fou d . e two horizontal intersections will determine the horizontal trac e ;

and two n n n n n the fro t i tersectio s will determi e the fro t trace .

t us no w n t b Le . I n Fi apply this method to a co cre e pro lem g. 9 the horizontal and front projections o f a triangular pyramid are

an w e wi sh to sh w th e l ane o f a s de a b c b its t shown , d o p i y races.

The projections o f two lines a b and a c in this side are marked

n n in the ﬁgure . The horizo tal and front in tersections of each li e

n n h and i with the pla es of projectio are marked respectively i f . The horizontal trace O f the plan e is Obtained by join ing the two

n n n and n b n n horizo tal i tersectio s , the fro t trace y joi i g the two

' i fron in tersect o ns . a b c n t If we call the side the pla e P , the

n be H n n F horizo tal trace should marked P a d the fro t trace P . The accuracy o f the drawing should be checked by O bserving

s n o n whether these trace i tersect the X axis .

I n n n n n n the example just explai ed , the li es determi i g the pla e

’ interse cted each other ; but the solution is the same for parall el r

A n n b an t ee n s no t in lines. pla e is determi ed also y y hr poi t the

n o r b a n an d a line . same straight li e , y poi t The ﬁrst case , that

n b n n b of the three poi ts , is most easily solved y joi i g them y straight “ ” I n th n n and and . e lines two two seco d case , that of the poi t

n b n h n an n i ntersecti n li e , solve y drawi g throug the poi t y li e g the

n B b th n n n given li e . oth ases are p resolved i to that of drawi g a

n n in es plane through tw o i terse cti g l .

1 b draw n a an e th r h o n e n e a el 9 . The pro lems of i g pl oug li par ll

n h e o r th r h a en n a e to tw o i en nes to a ot r , oug giv poi t p rall l g v li , are

n I n scarcely more than variatio s of this same problem . the ﬁrst

o ne n in n n b ns case , li e the required pla e is give y its projectio

n an n in . and the direction Of a other . Through y poi t the ﬁrst line

n n and b draw a li e parallel to the seco d , the pro lem is easily

I n n n n and solved . the last case the directio s of two li es are give , the - plan e is determin ed by drawing two lines parallel to them

n B i n through the given poi t . oth cases are resolved aga into

n n passing a plane through two i tersecti g lines .

s mee in i t I f w l in s intersect th e h o r z cti n li ne t a o n . t o e o n tal I nterse g p , i , in r fro nt and side pro j ectio ns tersect in th e co rrespon ding p o j ectio n s o f the , po int . li nes h a e th eir h o ri z o ntal fro n t an d side ro ectio n s arall el TP arallel v , , p j p ; r ll el th e tw o h o ri z o n tal the tw o front an d th that is if tw o l in es are pa a , , e , r ll el tw o side pro j ectio ns are p a a .

M 5 To ass a lane throu 2 0 R . h two s . P OBLE p p g i nter ecti ng or

r e s pa all l li ne .

e od F n n n n n n M th . i d the poi ts of i tersectio of the give li es with

n n n the horiz on tal and fron t plan es . The li e joi i g the in tersectio ns

z n n is z n n in the hori o tal pla e the hori o tal trace of the pla e . The

n n n n lin e joi ing the intersections i the fro t pla e is the fron t trace .

n an The side trace is located by the horizo tal d front t races .

E XE RCIS E S

n se a n n an d In the fo llowi g exerci s m rk the horizo tal , fro t ,

n n o t w d n d b e side trace s o f a pl a e other ise esig ate , y the l tters

P P an d . H P , , S P

— — “ n a 2 and b 1 19 . The li e through , , i

n u c - 2 0 and d mtersects the li e thro gh , ) — e — l 1 n at % , , Pass a pla e through

these two lines . D n O f an n i ntersecti n 20. raw the projectio s y two li es g at the

— 1 n n in poin t m , Determi e the pla e which

these two lin es lie . 1 D o f n parallel n 2 . raw the traces a pla e through the li es — — - — _ — — 1 b 2 mi I l , a i , ) , 2, g, t) ; a d i , i

—1 d — 1 - ) , i , n a — 2 — n : , i , 22 . Pass a pla e through the three poi ts

— w — ‘ — 1 — 1 — and 0 i ) b , , t) , t, t, r — n d — 1 F n n n , 1 2 3 . i d the pla e passi g through the li e — oint - 2 1 e and the p f § , h — 1 (a) Through the line g , pass a plane parallel to the line i — i ( — ~i i ) h b (b) Through i j pass a plane parallel to g . (O serve

that the traces O f the two planes are parallel . ) — 1 n n 0 , pass a pla e parallel 25 . Through the poi t — — and It 4 1 l 2 i to the lines g, ) , , — — — 1 n 2 2 m , , ,

Pass a plane whi ch is parallel to the X axi s through the

‘ n — 2 — 1 ne poi t p , , Draw the trace of the pla n o n the side pla e . b n s A skew bridge is shown in the ﬁgure elow . The cor er of the portal Of the b ri dge are given by the poin ts a

— — w — 4 - 2 b 2 2 c a , i . i . s, a ) , ( s, it rta d 0 as in . n ( , the ﬁgure The po l is the pla e d n b n a b c d and b . i n a b bou ded y the li es , , The l es and c d are called the en d posts and b d is the portal

ru n n o f a . st t . D etermi e the pla e the port l

b n o n The end posts o f the portal Of a skew ridge are show — — ’ in r 2 an engineer s drawings by the l es , i , — — — — - u 1 a v and t i s , , } , s l l a , i > ; H i , , The horizontal projection shows the plan and the

n D o n n front projection the elevatio . raw the three pla es

Of projection the traces o f the plane in which the lines

in an b n r s and t u lie . Measure degrees the gle etwee

i z n n . the plane Of the end posts and the hor o tal pla e b — 2 c The points a , — n d d 1 determine a surface . a ,

n n ? Can you ﬁnd a plane which will i clude these poi ts ? Is it then a plane surface — 2 b n a . A triangular pyramid is given y the poi ts

— 1 —u rn o f 2 7 and c at the co ers b ( 2 ) , b its base and its vertex d which is 1 % in ches a ove the , F n a b and 0. base and equidistant from , , i d the traces

f h n s o f a d b and b d c k n n O t e pla e , mar i g them P a d Q

respectively .

2 1 . I n b ems o f s ri i t are practice , the pro l de c pt ve geome ry mainl y those o f ﬁnding the intersections o f planes with other planes (Problem the intersections o f lines with planes (Prob lem o r the true lengths o f lin es (Problem These may be

n o f n n b o f spoke as the fu dame tal pro lems descriptive geometry .

n s no w n n so n Every stude t mu t lear these solutio s , that he ca use

n an n ns them immediately u der y co ditio .

F 1 a n and are n b I n i . c A g 0 two pla es P Q show y their tra es. n is n b n n tria gular pyramid also show y its projectio s . The pla e o f its side a b d is the plane marked P ; and another side b d o

n b s n i n o f w is the pla e Q . O viou ly the i tersect o these t o sides n d b o f i and no n i n is (pla es) is the edge , , the pyram d ; co struct o

n n b and d n b n necessary whe the poi ts are give y their projectio s .

o f ' n n b n n b s The ﬁrst the fu dame tal pro lems me tio ed a ove arise , n n b and d not n n s however, whe the poi ts are k ow , the trace

f n n and n in n in o the pla es are give , the li e which the pla es

n b tersect must be fou d . The pro lem to be solved is shown

F l b the H F nd F in i . o a more simply g , where traces P , P , H Q , Q n i n n nn are show without the pyram d . Expla atio is u ecessary to show that if a line in the plane P in tersects a line in the

n a in in th e n e o f n tersect n o f th e tw o an es pla e Q , po t li i io pl

s determin ed . z n is ine in n i The hori o tal trace , H P , a l the pla e

z n n e in n P ; and the hori o tal trace H Q is a li the pla e Q . Where and n n n i n i n o f n H P H Q i tersect is a poi t , the , the requ red li e i ter n n e section Of the two pla es . The horizo tal proj ction o i this point

" d n n nf n in is n an . , the fro t projectio , , is evide tly the X axis Sim

il arl n F and F n of the y , where the fro t traces , P Q , i tersect is , n n n in n o f n n front projectio of a other poi t the li e i tersectio . The k z n n o f n is o in i hori o tal projectio this poi t the X axis . The hor

z o ntal and front projection s Of the line o f intersection are found h h n o n rd of n then by drawi g n a d . If these projectio s are drawn

F l a can o f u r in i . 0 o b Observ also g , we check the accuracy work y

n o in e d b it shoril d ing whether co cid s with as .

2 2 I n b s n . some pro lem where the li e o f intersection Of two

n to be n o f n not n pla es is fou d , the traces the pla es do i tersect

o n n o n the sheet which the drawi g is made . Such a case is shown

1 1 n no t n in i n Fi . i g , where the fro t traces do i tersect the lim ts of

w n Fo r n o f b an n A the dra i g . the solutio this pro lem auxiliary pla e , , parallel to the front plan e o f projection is drawn to in tersect the

n nd z n a . n n b A pla es P Q The hori o tal trace of this pla e is show y H . hi ne n n and T s auxiliary pla cuts auxiliary li es from the pla es P Q . n n o f n n n The horizo tal projectio s these li es are , Of course , coi cide t

A and n n with the trace H , the fro t projectio s are parallel to the

n n n of o f n n fro t traces . The i tersectio , , the fro t projectio s Of these auxiliary lines determi nes o ne point in the line Of intersec

‘ h n and the ho ri z o ntal n o is of i n tio , projectio , , vertically over

ri n r ec n f n A s n in the ho zo tal p oj ti o s o both li es . explai ed the pre

n n n n in n o f n cedi g paragraph , the poi t is also a poi t the li e i ter h h n o f n n B n n n o and nf of sectio the pla es P a d Q . y drawi g the b n n i n f n n we O tai the two projectio s of the required l e o i tersectio . If i n this problem the hori z ontal traces did no t intersect o n the

an i n be n paper , aux liary pla e would eeded parallel to the hori z ontal n pla e .

2 n n 3 . I a case where both pla es are parallel to the X axis their line o f intersection is likewi se parallel to the axis ; and the requrred lin e is determined by the intersection o f the side traces

2 n and re re o f n n I n Fi . 1 the give pla es . g the pla es P Q are p n to i sented by their horizontal and fro t traces parallel the X ax s . The side traces o f the planes are shown at the right o n the draw ” i z n nd n o in n n n as . r a r , i tersecti g at the poi t The ho o tal fro t g p"" j ecti ons o f the required line o f intersection are shown by x x 7 " and x at .

two lanes d he li ne o i n ersecti on o . 2 R M . o n t t 4 . P OBLE 6 T ﬁ f f p

e n ri z n n is the M thod . The poi t where the ho o tal traces i tersect

n f n n o f n ri z n n poi t o i tersectio the required li e with the ho o tal pla e . The poin t where the front traces intersect is the poin t o f intersec

n n A n tion Of the required line with the fro t pla e . fter locati g the two n o f o f n n projectio s each these poi ts , joi the like pro jections

n f n n to Obtain the li e o i tersectio .

E XE RCISE S

° F n n o f n n o f two n — 31 . 3 i d the li e i tersectio pla es M ,

and N — 1 ,

2 n and n n o f in f 3 . Draw the horizo tal fro t projectio s the l e o intersection o f the plane of the end posts in Exercise 2 8

ario ther n to n an with pla e , parallel the X axis , maki g ° angle o f 30 with the hori zontal plane and passing through a point 1 inch behi nd the front plane and l in ch below

n n n the horizo tal pla e . (Use the side pla e . )

F n n o f n n o f n 3. 3 i d the projectio s the i tersectio the pla e , M ,

in 31 n — 1 Ob Exercise with the pla e P , ( serve that the li ne o f intersection is parallel to the front plane and that the fron t projecti on i s parallel to the t front races . )

4 F n n O f n n O f n 0 3 . i d the li e i tersectio the pla e Q ( , with a vertical plane parallel to the front plane and 1

n n i n n n i ch behi d t. (The fro t projectio of the li e o f

n n n Th i tersectio is parallel to the fro t trace of Q . e r o z l n ho i nta projectio is parallel to the X axis . )

5 F n in O f n n o f n in 1 3 . i d the l e i tersectio the pla e , M , Exercise 3

in 4 n 3 . an li with the pla e , Q , Exercise (Draw auxi ary n nt n n n pla e parallel to the fro pla e , cutti g the pla es M

n n and Q . It will cut from each of these pla es a li e n n n n parallel to the fro t pla e . The i tersectio of the two li nes thus obtained is one poi nt in the required line o f

intersection . ) — — n a b 2 1 the n o f 36. Take the li e , , as li e

R an I an n of two n d . n d n intersectio pla es S y irectio ,

n n a n t through the fro t projectio of , draw the fro traces

n n the n o f the two pla es . Determi e horizo tal traces .

13 n and n o f b I n Fig. the pla elevatio s the roofs of a sta le

m for the ho ri z ontal r o ection an d elevati on f P lan i s an other na e p j , or ns the fron t o r side proj ectio .

and an n n adjoi i g shed are shown . Complete the draw ings by showing the projections of the lin e where the

n n n n two roofs i tersect . The dime sio s a d slopes neces

n traces n sary for drawi g the of the roof pla es are given .

b and R and Mark the sta le shed roofs respectively S. Take scale

Su esti on —I n the ure th e horiz ontal l an e o f ro n gg . ﬁg p p j ectio i s taken throu h he o ttom o f the shed ro o f thus l acin the X a is as it is show n g t b , p g x in the draw in Th fro n t traces O f the ro o f l an s F nd F r . e e R a S a e g p , , , then co in ciden t w ith the lines sho w in the ro o f sl o es in th e front ro ec , g p p j tio n Th h ri z o n tr ces H R an d H S are draw n thro u h the i nter . e o tal a sec , , g tio ns o f the fro nt traces w ith the X a i s and arall el to the l in es in the x , p pl an w hich sh ow the intersectio ns o f the ro ofs w ith the h o ri z o n tal pl an e o f

’ ro ectio n The l in e Of intersectio n Of these tw o l anes R and S deter p j . p , , — min es the mpe O f the line O P the l ine sh o w ing the i n tersecti o n s o f the ro o fs n h l an i t e p . rs c i e raw in Fi 1 4 is added merel to m a e the o ther The pe pe t v d g ( g. ) y k r w s l ainer d a ing p .

n n b r n n 2 5 . Of the fu dame tal pro lems , the ﬁ st , that of ﬁ di g the

n n n be no w in intersection of a pla e with a other pla e , should well

l no w n n t mind . We shal take up the seco d of these most importa

n n n n n n problems , that of ﬁ di g the i tersectio of a give li e with a

n be n an h er an e —‘ given pla e. It should plai that if ot pl , called

n —i s assed h h th e i en n e th e usuall y an auxiliary pla e , p t roug g v li , poin t wh ere th e given line intersects th e given plan e i s in th e lin e

n e h th e en ane o f intersection o f th e auxili ary pla wit giv pl . To

n n bo a d . o ne illustrate , take two sheets of card ard a pe cil Place sheet o n a table and hold the pencil in any way to intersect this

' ~ soli that o ne sheet . Now hold the other sheet of its edges touches i m n n n . the ﬁrst sheet , while at the sa e time the pe cil lies its pla e Observe that th e pen cil n o w touch es th e sh eet o n th e table i n th e

' h s n li ne o f intersecti oiro f th e tw o s eet . This same pri ciple is

i H n 15 . to shown also in the pictorial drawing in F g. ere we wa t

n n show the intersection of the line 0 d with the pla e Q . The li e h h n c d and of df and n b is shown by its projectio s , , the pla e y its n A b and F . The auxiliary pla e, , marked y its traces , H Q Q

A and F A n c d . n traces , H , is passed through the li e The pla es

"h and A int in n n b n i n n Q ersect the li e represe ted y the projectio s , Remember th at th e oiri t wh re th e li ne c d inter and ml 111 . p e

h l an e must be in th e l n e n sects t e p Q i m . This required point is in n 0 d of . n n n also , course , the li e We k ow , the , that the poi t we

n in b O f n n and c and are seeki g is oth the li es m d, therefore at

n i n . the poi t , , where they i tersect

n in n in Fi The same problem is show orthographic projectio g.

n 0 d i n b n and n b 16 . The li e s show y its projectio s the pla e Q y

u l n A n 0 d ts traces . The a xi iary pla e , , is passed through the li e .

n n a an n n n in The solutio is ge eral , so th t y other pla e i cludi g the l e

n li n F n n 0 d could be take for the auxi ary pla e. ewer li es are eeded

n w r n is n er en for the drawi g , ho ever , if the auxilia y pla e draw p p

o f n o f I n Fi 15 l r o ne n . dicu a to the pla es projectio g. it is per

endi cul ar n n so n A p to the fro t pla e , that the horizo tal trace , H ,

n . n and A n t is perpe dicular to the X axis The pla es , Q , i tersec in the line m n ; and the point where the line c d intersects the

n nt n n and c d — at n i n pla e Q is at the i ersectio of m , _ the poi t , show h n and n e ns i and 17 I n by its horizo tal fro t proj ctio . this partie h ul n i mi n b n n o ar case , the projectio is deter ed y the i tersectio f

n o f m n d n and e . n n the horizo tal projectio s The fro t projectio , h 17 n b n n n r i to n , is fou d y drawi g a projecti g li e th ough i tersect

nt n o f n and c d n the fro projectio s m , which are coi cident .

To nd the 2 6 7 . oi nt n w c . PROBLEM ﬁ p i hi h a gi ven li ne i nter

e sects a gi ven plan .

' od an n n Meth . Pass y pla e through the give line tusrﬁi lly a n n in e f plane perpendicular to the fro t pla e) . The l o intersecti on

f n n n n ts o this auxiliary pla e with the give pla e , i tersec the gi ven i line at the point that is requ red .

EXERCISE S

F n n n — k —1 i d the poi t where the li e j 1, 5, — 1 n n U 0 ) i tersects the pla e ( ,

Determine where the line j lc given in the preceding exer

n n W 0 c ise i tersects the pla e ( ,

Draw the plane P Find where the lin e a — b — 2 n 0 . i ) , ( , , i tersects it

L n n c —1 — d — 1 ocate the poi t where the li e § i ,

— 1 n o f b , passes through the pla e Q which oth the n n n . a d horizo tal fro t traces are parallel to the X axis , and are respectively 55 inch behind an d i i inches below

this axis .

42 he n in n n r . The top of t desk show the accompa yi g ﬁgu e is — b n a — 10 8 b 12 located y the poi ts , , ,

" — - c 0 —6 and d 4 2 A ( , , , , light placed — l - 14 10 2 has so at , , + ) its rays reﬂected that the ° n n is n n 6 ver cal most i te se light i cli ed 0 from the ti . Show

O f n n o f of inten the curve i tersectio , the rays maximum

sit n ‘ y with the pla e Of the desk .

43 n n r F n . The pla e M represe ts a mir or . i d the point where a ray o f light passing through the points

— — - 2 1 and s r , ) 5) is reﬂected from

the surface of the mirror .

- A in b n n . 44 . steam pipe a uildi g passes through a slopi g floor The axis of the pi pe i s located by the points v — and w 1}1 The plane of the ﬂoor may be represented by the plane F Find

n n o f o f the i tersectio the axis the pipe with the ﬂoor .

2 Lines h at are aral el to a an e a ear in th e r tru e en h 7 . t p l pl pp i l gt

ns n h n I n Fi 1 7 n and in th eir proj ectio o t at pl a e . g. the horizo tal L n front projections o f a hip roof are shown . i es that are parallel

n n o r n n either to the horizo tal pla e to the fro t pla e , such as the

in n n o d c e in n . li es or the ﬁgure , are here draw their true le gth

n b b n n A li e that is o lique , however , to oth pla es of projectio , as

or n b is no t n in n in f example the li e i , sho w its true le gth the drawing ; bu t in this case a thi rd view may be made to show its

n n is in t- n n in true dime sio s . This illustrated the righ ha d drawi g the

o n n V n ﬁgure . The roof is here projected a vertical pla e , , draw

n b and n n n through the li e f, the this vertical pla e is revolved i to n n n n by the pla e of the drawi g . This projectio is co structed draw

’ ’ ’ At o n in b n V V V V . n g the ase li e , , parallel to g the projecti g h ’ ’ ’ ’ ’ b o ff n n b b n V V . li e produced , the li e g is laid perpe dicular to

n n b o r Its le gth is equal to the altitude Of the poi t , to the

’ ’ f f in n n distance b g the fro t view . The li e b f shows then the n O f n true le gth b f . The completed drawi g shows also the true n n a c to n V . le gth of the li e , as it is parallel the vertical pla e ,

2 8 I n n n . practice these co structio s are simpliﬁed by passing the h h n t ro h auxiliary vertical pla e h ug b i . The line V V is the trace of

n in n this pla e the pla e of the base Of the roof . If this plane i s

v n b b V V n re olved , together with the li e i , a out to coi cide with

n o f b n b l n in n e the pla e the ase , the li e f wi l the lie a pla e wher n n can e true dime sio s b measured .

The revolution o f o ne line about another line as an axis is ao

ed b n n in the ne A complish usually y the revolutio of two poi ts li . point revolved thus about a line as an axis describes a circle with a radius equal to the actual perpendicular distance from the poin t

I n in b and be b to the axis . this case the po ts f must revolved a out in n o f b so the axis V V . The axis is here the pla e the ase , that the perpen dicular distances o f the points from it can be measured in

Fo r n b s n f bf the fron t view . the poi t this di ta ce is g , which is laid h O ff o n n b n l V V . the li e through , perpe dicu ar to The revolved ” i in r ol n b b . n s in ev u positio of is at The poi t f the axis , so that ” h n b is n n b n tion it is stationary . The li e f the the positio Of f whe n n n and n revolved i to a pla e of the drawi g shows its true le gth .

to a o a ven st n e o 2 9 . If it is required l y ff gi di a c n a line no t shown in n in n n b its true le gth a drawi g , we must ﬁrst revolve the li e a out an axis to bring it into a plan e where its true length is shown and

Fo r n n . the measure the dista ce example , if we wish to measure a

n Fi 17 n n b 33 n b . b dista ce , , alo g the li e i ( g ) from the poi t , we lay ” h o ff n n o n n b n the give le gth the revolved positio f , locati g the

n x and n n n at poi t the reversi g the precedi g process , we revolve h h n n b n n o f b x i back i to the li e i . The horizo tal projectio s then b sc

0 . n n n to be n it 3 Usually whe the true le gths of li es are fou d , is most convenien t to revolve them into either the horizontal or

n n n n an n fro t pla es , rather tha i to auxiliary pla e which is parallel

n n n to o ne Of the planes o f projectio . Of course a y li e which lies ' i n o ne n o f n is wn in n of the pla es projectio sho there its true le gth .

f n in Fi 1 a n o . 8 This method solutio is illustrated g , where a li e m n n i t in and b n is show as is located space , also y its , projectio s "h i i n in in n in i n n and m n . The poi t this li e lies the horizontal h n so n is nl b n m i pla e , that this poi t marked o y y its projectio . Th s ﬁgure shows the line m n also when it is revolved in to the planes

n Let n it Of projectio . us co sider ﬁrst how is revolved into the

z n 777. in n . n b n n in r hori o tal pla e The poi t ei g already this pla e , evo l i n n n n the n n ut o remai s statio ary . O ly poi t therefore must be

' h n o f n n revolved to s ow the revolved positio the li e . This poi t h moves in the arc of a circle about its horizontal projection m as a

n and n n n b ce ter , with a radius equal to the dista ce the poi t is elow

’ n n n the horizo tal pla e . Its revolved positio is at n I n a h n n is n b n n n drawi g , however , this revolutio show y co structi g at

h " in n ul a to the i m n and n o o n a l e perpe dic r ax s , , layi g ff the per

endicul ar t n n n b n n p the dis a ce the poi t is elow the horizo tal pla e .

B i n n n in n s y the d me sio li es the ﬁgure , this dista ce is marked . h ' n m n n is n in n The li e is thus obtai ed . It a li e the horizo tal

n and n in n I n pla e is , therefore , show its true le gth . the same way the true length Of the lin e w as found by revolving into the

n n in m and n n fro t pla e . The po ts revolve ow respectively to

’ ’ an n n n n n t n u m d n in the fro t pla e . Dime sio li es a d show the i n s n t s . n n dista ce that , hi case, are equal The true le gth is show ’ ’ m n n n n n b . y If the li e m is revolved i to the side pla e , the same

n result is obtai ed .

F 1 a all i inl in igure 8 shows th s very pla y as a picture . This

is for b in ro method , however , the same pro lems orthographic p

c i n I n n n is n n i 1 b t o . i F e . 8 j this way , the same li e m represe ted g b n and n n and n n y its horizo tal fro t projectio s , its true le gth is show h ’ ’ ’ now m n and m n n l as very accurately by . The stude t shou d me

r n a u e these le gths to satisfy himself that they are equ l . A case occurs sometimes where the horizontal and front planes

no t b n n n h have ee adva tageously located , so that li es pass throug

n n ro and continue beyond the planes of projectio . The the p jec tions o f parts o f the line fall o n Opposite sides of the X axis . When

in n n it a such a l e is revolved i to the pla e which p sses through , distanc es must be laid off o n opposite sides o f the axis o f revo i l ut o n .

i 1 n a c is n n n and a n r I n F g. 9 a li e show i tersecti g p ssi g th ough

n b n n n the horizontal pla e at . If this li e is revolved i to the horizo tal h h a c as an n b n n and plane about axis , the poi t remai s statio ary ,

os te d re ons n ai a and c revolve in opp i i cti . The dista ce that is below h h a o n o ne o f a c and n t the X axis is me sured side , the dista ce tha

‘ ' ’ o n n a c a is above is measured the other side . The is the true

n n th f n . e length o the li e If , however , the same li e is revolved i to " ’ n no t as a c now front pla e , this difﬁculty is met , shows the true length .

To nd the true len th o a li ne i ven b its 1 R E 8 . 3 . P O BL M ﬁ g f g y

he d sta e between two o nts projections; or to ﬁnd t i nc p i .

an n r ndicu Method . Pass auxiliary pla e through the line pe pe lar to a plane in which lines are shown in their true lengths o n the

n f n n n drawing (usually a pla e o projectio ) . With the li e of i ter

n n as an n n n sectio of the two pla es axis , revolve the li e i to the seco d

n n in n pla e , where it is show its true le gth .

E XERCISE S

F n n b n n a — 5 l and i d the dista ce etwee the poi ts , ,

b — l - 3 , ,

F n r n b n n 0 —6 — 1 —2 i d the t ue dista ce etwee the poi ts , , )

and d — 2 ,

L in n n n n ocate , a pla e parallel to the fro t pla e , two poi ts that are two in ches apart and are no t equal distances

n n below the horizo tal pla e .

A n s n e — 5 0 and l li e pas es through the poi ts , , f ( ,

D n n F n raw the side projectio of this li e . i d the length o f this line included between its intersections with the horizon tal and front plan es ; and with the front

n and side pla es .

Two stations 3 1 and 32 are to be connected by a telegraph

n 3 —3 and 3 line . The locatio s are 1 , 2 Find the length O f the shortest line that

n will connect the two statio s .

b nn n n . 50. Fi d the le gth Of the shortest elt to co ect two pulleys B oth are two inches in diameter and are i n the same — n b 1 and plane . Their ce ters are at , — c i ,

1 n its n A line is 4 3 inches long . The projectio s of e ds are — b n a - 5 1 and b - 1 — 3 located y the poi ts , , , ,

n n Determine the projection that is o t give .

in n - 3 One n A line lies the pla e M , projectio — o f the l rne i s given by the poin ts g 1% and — 1 O n n O ff 1 7 n h , this li e lay } i ches from its

n n intersection with the fro t pla e .

2 Lin es P er endi cul ar to a Pl ane . T o n 3 . p w pla es P and H are

A n a c is n in i . 2 wn F 0. n sho g li e perpe dicular to the pla e P . h From the point a the proj ecting line a a is drawn perpendicular

H n n h to the plane . The pla e determi ed by the lines a c and a a H n n and . er en is perpe dicular to the pla es P It is , therefore , p p dicular in n o f n to the trace H P which is the tersectio the pla es . h " n a c n n in h n n The , which is a li e lyi g t is pla e , is perpe dicular to

ns now n n n H P . Co ider the pla e H as a pla e of projectio ; then h h a c n O f n a c of h is the projectio the li e , while H P is the trace t e

n wn n n n a c is n pla e P . We have sho , the , that whe a li e perpe dicular

n n n n to a pla e P , its horizo tal projectio is perpe dicular to the hori

n f f r z ontal . o o r trace The same relatio holds , course , the othe

on of n and n n of n projecti s the li e the correspo di g traces the pla e .

I f h en a stra h i ne i s to be drawn en d cu ar to , t , ig t l perp i l a

l e it i s n necessar to draw i ts ec ns er en di cu ar p an , o ly y proj tio p p l

es o f th e an e An d to d aw to th e co rrespondin g trac pl . r a plan e

n u ar to a ine draw th e races er en d cul ar to th e perpe di c l l , t p p i h l n projection s of t e i e .

To nd the di stance rom a t R M 9 . i ven oi n to a 33 . P OBLE ﬁ f g p

e gi ven plan .

e hod D the ven n in er endi cu lar M t . raw through g poi t a l e p p to

F n n n n n n . the plan e . i d the poi t where this li e i tersects the give pla e “ The required distance is the true length o f the li ne joining thi s last point with the given point .

EXERCISE S

A n — 2 n to in n pla e Q , is perpe dicular a l e draw n 2 through the poi t j , Draw the projections

n n Of the line ; and ﬁ d the dista ce from j to Q . 42 ﬁnd n the n t ne I n . 54 . Ex the dista ce from poi t to the pla f Of the top O the desk . Find the distance from the poin t w in the axis of the steam

i 44 n O f pipe given n Ex . to the pla e the floor .

n b n 0 A A hillside is represe ted y the pla e S ( ,

pole is planted o n it perpendicular to its plane . The b n t t — l ottom of the pole passes through the poi i ,

— - 1 n as th 1 . b , i ) Measuri g y the same scale for e

o n t n 1 n b the co rdi ates of , locate a poi t that is i ch a ove

n n O f o f i grou d . What is the le gth the part the pole n the ground ? A poin t i is above the roof represented

n R 0 A n n by the pla e ( , shaft for tra smitti g “ power to another building passes through this point

L n n perpendicular to the roof . ocate poi ts o the shaft

an 2 n 1 inch above the roof d i ches below . Draw the projections Of a cube and ﬁnd the distance from

n n r an n n a y cor e to oblique pla e passi g through it . The

n n t e an pla e should o b parallel to edge .

F n n b n n of i d the dista ce etwee two parallel pla es , which

all the traces are Oblique to the X axis .

To ro ect a i ven li ne on a e e R 10. u i v n lan 34 . P OBLEM p j g p g p .

hod an tw o n in n an et . d M Select y poi ts the li e through each ,

n n n n The n draw the projectio s of a li e perpe dicular to the pla e . li e joining the poin ts where these perpendicular lines intersect the given

n f n n t n plane is the projectio o the li e upo tha pla e .

EXERCISE S

P n - 1 - 1 n —1 n 60. roject the li e m , ) upo

n — 1 the pla e P ,

The major and minor axes of an elliptical cam- wheel are

n b n d 1 determi ed respectively y the li es 4, — e — 1 1 — l — ( t, t, i , i ,

Draw the projection o f this cam - wheel o n the plane

’ T — 5 b n n b O f , y projecti g a um er diameters

o f the ellipse . The vertices Of a triangle shown in the ﬁgure are at

_ _ the in : a ’ b 1 _ l ' c po ts i 1 ( Fr i )

t s n n the ane R (0, Projec thi tria gle upo pl

ne n is n b n and an 35 . If o li e give y its projectio s , a pl e is to be

n n n n in Ar draw perpe dicular to it , it follows from the expla atio t. 32 that the traces o f the required plane must be drawn perpendicu

n n n O f n n I . n F lar to the correspo di g projectio s the give li e igs .

2 1a and 2 1b n a b n b tw o n and r a li e is show y its projectio s , it is e quired to draw through a given point 0 a plan e perpendicular

n 0 w n m 0 to the line a b. Through the poi t dra a li e parallel to

n n in n the horizo tal pla e which shall lie the required pla e . A line which is parallel to the horizontal plane is commonly called “ ” n and n n n a horizo tal , is a li e with its fro t projectio parallel “ A nd n in to the X axis . if this horizo tal shall lie the required

n z n n be n pla e , its hori o tal projectio must parallel to the horizo tal I n in O f n to be n . trace the pla e draw other words , this case , the conditions to be satisﬁed are : ( 1) that the front projection W of ” of the horizontal must be drawn parallel to the X axis ; and (2) h h that the horizontal projection m o must be drawn perpendicul ar

— to the horizontal projection of the gi ven line a b the same as saying

t n an it mus be parallel to the horizo tal trace Of the required pl e .

H n n n m o n in n avi g thus determi ed a li e lyi g the required pla e ,

n F n n n i of the fro t trace , P , is draw through the i tersectio (f ) n n an d n bf n m o a) . the li e with the fro t pla e , perpe dicular to The

n n n n F horizo tal trace , H P , is draw through the i tersectio of P h h b A n r and n a . e en with the X axis perpe dicular to pla e , P , p p

di l r n a b and n n n 0 cu a to the li e passi g through a give poi t , thus ‘ is

n i determi ed . A line parallel to the front plan e and i ntersecting the horizon tal n e n n n O f be pla e (d termi i g the horizo tal trace ﬁrst) might , course , “ ” i n o f z n b in . used the place the hori o tal , to O ta the same result

M To ass a lane throu h a i ven oi nt er 6 R 1 1 . en 3 . P OBLE p p g g p p p

c lar to a ven li ne di u gi .

e ho D w n n n M t d . ra through the give poi t a li e which will lie “ ” in the required plan e (a horizontal lin e is usually most easily

n n n one n used) . Through the i tersectio Of this li e with of the pla es o f projection draw o ne trace Of the required plane perpendicular

n n n o f n n to the correspo di g projectio the give li e . The other trace is drawn perpendicular to the corresponding projection o f

n and n n o f the li e , through the i tersectio the ﬁrst trace with the X axis . EXERCISE S

— — u n e 1 1 n U Draw thro gh the poi t , , a pla e , ,

be n n — 3 — l which shall perpe dicular to the li e g , } , — —1 h “ ) , ( Z i ,

n b 0 and 0 — 5 in The poi ts , , are the

n eaves of a roof . The poi t g is in the

D n w . 0 ridge raw through the poi t ( , the

proj ecti ons O f an arrow that will show the direction o f

n on a force due to wi d pressure the roof . Make the tip

o f to gh the arrow u the surface Of the roof . (Pressures o f ﬂuids are perpendicular to the planes on which they

are exerted . )

D n to n 0 a raw a pla e parallel the pla e P ( , at

n n o n th - n dista ce of two i ches from it e right ha d side .

I n n n n 37 . a complicated drawi g it is ofte ecessary to ﬁnd the true size o f the angle between lines in a plane surface which is no t an n n n n parallel to y pla e showi g true dime sio s . The usual process is to imagine the plane Of the surface extended to in tersect the plan es

o f n nin n and n projectio , thus determi g the horizo tal fro t traces

f n in Ab t o n f o the pla e which the surface lies . ou e o these traces

an n n n as axis , the pla e is the revolved till it coi cides with the

z n o r n n hori o tal fro t pla es . The simplest case of ﬁnding the true size o f a plane surface is

- n n in Fi 22 illustrated at the left ha d side of the roof show g. . The

a b c no t n in n n side is show its true size , either are its a gles . If ,

b n n n however , the side is revolved a out its i tersectio with the pla e h h o f b b c n n b the ase of the roof , , i to the pla e of the ase , its true size

n n n a b and a c n in will be see . The the li es will be see their true

n and th e true s z e of th e an e betw een h e I n le gths with i gl t m . this h h n b a b b n a revolutio a out the axis , , the path traced y the poi t is shown in its front projection by the dotted lines of the arc o f a

' f f d n o n b . a circle through , with its ce ter at The poi t shows its ’ h h n in n o f b and a b c revolved positio the pla e the ase , shows true

n an n dimensio s for y part Of the outli e of the surface .

' 8 n n b b 3 . I the precedi g case , the pro lem was simpliﬁed ecause the surface to be determined was perpendicular to the front plane

o f n n c n a projectio , so that the radius for the revolutio of the or er

n in n n A w as shown in its true le gth its fro t projectio . t the right

f i 2 2 o f d e n o F . n ha d side g , where the side the roof f is represe ted , n H n a more difﬁcult problem is prese ted . ere whe the revolution h h is made about the line e f (the intersection o f the side wi th the

n o f bas as an n d in pla e the e) axis , the poi t will move the arc Of a circle wi th a radius equal to the perpendicular distance from d to

b n d is s . e e the axis . O viously the li e g this radiu The p rsp ctive

’ nd b o n d d w in Fi . 2 3 a vie g shows this radius , y the arrow the

n tru e n * n a directio n of revolutio . The le gth is fou d e sily from its

T h e true l ength i s measu red by th e hypo th enu se O f th e righ t tri angl e h h h d d g i n th e h ori z o ntal pro j ection . I n this tri angl e d d i s l aid off equal to f f f f f the distance f rom d to th e l ine of th e base o f the roof b o c f .

f n l - le f . n d o this two projectio s The true e gth , g , radius whe n laid o ff o n h h ’ d r n n d d g produced , shows the evolved positio Of the cor er , , at . ’ h " O f giveri n b n d e The true size the surface is show y the tria gle f , m n can be fro which true a gles measur ed .

ers ecti ve /view o f n in 2 The p p this roof represe ted Fig. 3 shows h h more plain ly the right triangle called d d g i n the preceding con

on t r are b b n i re o structi , ogethe with the descri ed y the poi t d n v lv ’ h h b e an ing to d a out f as axis . h T h e ri gh t tri angle d d g ( Fig. 23) sh o w s al sO th e true angl e th e side d e f

ma es w i th the l ane of th e base. Thi s an l e 1s mar ed a i n th e ﬁ u re k p g k g .

5 i : 24 sh ow s the sirn lest t n of b to ﬁn 39 . F g p sta eme t the pro lem d

e w een two n erse in lin es m n and o n th e true angle b t i t ct g p. No e Of the angles between the projections Of these lines show the true angle between the lines ; and they must be revolved into a plane

n B n n . n where true dime sio s are show efore , however , the li es can n o ne o f n n o f be revolved i to the pla es of projectio , the traces the

n plane in which they lie must be determined . The i tersections o f the lines with the horizon tal and front planes are marked respect

l h n n n n in n ive y i a d f i . The li es joi i g these po ts that are correspo d n n i ng are the traces o f the required pla e . The li es must then be revolved about o ne o f the traces in to the corresponding plan e of

n b t n projectio ; that is , we must revolve a ou the horizo tal trace

n o z n n b n t n n l n i t the hori o tal pla e , a out the fro t race i to the fro t p a e ,

I n n n n z n etc . the ﬁgure the li es are show revolved i to the hori o tal

n b n as an . n pla e a out the horizo tal trace , H P , axis The true le gth

f n n n n i o f o the radius for revolvi g the poi t of i tersectio , , the two lin es is found by constructing the right triangle shown at the right

n n n O f the drawi g . The hypothe use of this tria gle is equal to the

n true distance from i to H P . The revolved positio of i is shown

’ at i by laying o ff a distance equal to the length of the hypothenuse (

nd n m at right angles to H P . The e s of the li es n and o p are

I n racti ce u su al l thi s ri ht trian l e can b e l aced o n the d raw i n s p , y , g g p g o nl n l m b e ran sfe rre F r e am l n that o o e e u st t d . o e i s tead of co n structin y g x p , g the ri ht tri an l e as sh o w n th e distan ce mar ed a mi ht ha e een l ai g g , k g v b d O ff

‘ w ith di iders o n the X a is at ri ht an l es tO the distan ce m ar ed b an d the v x , g g k h o h nuse co ul b e measu red i n o n e o era io n yp t e d p t .

in n n n e n t na and . in H P; revolvi g the li es , these poi ts r mai s atio ry The lin es shown by heavy dashes in the ﬁgure represent the lines

n and o n t n n n m p whe hey are revolved i to the horizo tal pla e .

n 9 of n b n tw n s The a gle [ is the true size the acute a gle etwee the o li e .

b n n nl o ne n r O serve that for the co structio o y trace is eeded , the othe might be omitted .

M 2 To nd the e s e o 0 1 . tru i z the 4 . PROBLE ﬁ f angle between two

s nes i nter ecti ng li .

ethod in one of of n in M . Determ e the traces the pla e which

th e n n . an two give li es lie With this trace as axis , revolve the

n n o f n in n li nes i to the pla e projectio which the axis lies . The a gle between the lines in their revolved position is the true size of the

i n requ red a gle .

E XERCISE S

et n n o f n a b and b 0 i n th 66 . D ermi e the pla e the li es g ve by e

r in a — 2 b co b d ates : , and

c — l — n O f i , 1, With the horizo tal trace this

n as an n n n pla e axis , revolve these li es i to the horizo tal

n pla e . — n a - 2 b 1 — n s The li e , , 1) i tersect

n 0 — 2 d — l 6 the li e , %) at

1 F n o f b s n b n , i d the true size the o tu e a gle etwee

n these two li es .

A b n n b tw o l o ne n 68 . oat is towed alo g a ca al y mu es , walki g o n

n I n t n n b b . n each a k projec io drawi g , the oat is show at

b — 2 to w - n ot , The paths are at the same

and the o sitions of wn at m level , c p the mules are sho 1 — — — — 1 1 and m F n the , , i ) 2 1, i d true

n b n to w- size of the a gle etwee the ropes . If in the preceding exercise the forces exerted by the two b mules are the same , show the course the oat will take .

u esti on - S gg . If the forces in the two to w ropes are

e b is qual , the course of the oat shown by the bisec to r n b n Of the a gle etwee them .

of n n a —3 The vertices a tria gle are at the poi ts ,

b — 2 — 1 and c , i , Draw the projections o f the bisector o f the angle between the sides

a b and b c on n n and n the horizo tal , fro t , side pla es .

The o f is n b n c — 2 mast a derrick show y the li e ,

d — 2 - 2 - . Gu to g) , , i ) y ropes attached to the p o f the mast at c are fastened to the ground at a

—2 —2 and b — 4 — , ) , i ,

a Find i the n b n - a c and b c ( ) a gle etwee the guy ropes . (b) Find the angle between the mm b c and the mast d c .

Check this last result by revolving b 0 into the fro n t n pla e . h h u esti on h in S gg . Observe that the i for c d is at c d

he in t draw g .

1 n w a n a i i n n n 4 . The a gle hich li e m kes w th ts projectio o a y

n t n b n n and n pla e measures the rue a gle etwee the li e the pla e . By drawing a line perpendicular to the plane from a poin t in the

n n — n n i n er endi cu give li e a right a gled tria gle is formed , which the p p

o ne o n n n and n lar is leg , the pr jectio of the li e is a other leg , the li e

n n b n itself is the hypothe use . The required a gle is that etwee the hypothenuse and the leg lying in the plan e ; but the other acute

n o f n n n b n ne a gle this tria gle , the compleme t Of the a gle etwee the li

Ar t . 4 and n n b 0. the pla e , is more easily fou d y the method of

M 13 To nd the an le between a i ven li ne and a 2 R . 4 . P OBLE ﬁ g g

e lane gi v n p .

e ho an n in n n n er en M t d . From y poi t the give li e draw a li e p p

w o n n F n n b n t . dicul ar to the pla e . i d the a gle etwee these li es The complemen t of this angle is therequired angle between the line and the plane .

E XERCISE S

— — — 2 F n n e n a 1 1 — l b 7 . i d the a gl the li e , ) , 5

n - 1 makes with the pla e P ,

— F n n in c 2 — d — 1 — i d the a gle the l e , 4, , 1,

n — 3 b makes with the pla e M , (O serve that the angle most conven iently found in this case is the supplemen t o f the angle between the perpendicular and the given lin e as shown in the right triangle men

in Ar tioned t.

° The slope Of the bank o f a canal makes an angle of 30

- n o f n with the water level , so that the pla e the ba k is

n e b 0 as n in A r . eprese t d y P ( , show the ﬁgure

nk b t in n mule o n the ba draws a oa the ca al . The traces ° on the mule are attached to a rope at a po int m

—3 end n , The other Of the rope is faste ed to a — b - 4 l As n n o sa boat at , , sumi g there is g i n the rope (that the poin ts m and b in the ﬁgure are in a

n ﬁnd n b n an straight li e) , the true a gle etwee the rope d

f n the bank o the ca al .

I n n n o n of 75 . co structi g shadows we take rays light

n - n and that are reflected dow ward from the left ha d side , are shown by lines whose horiz ontal and front projections

° ° make respectively angles o f 4 5 and 315 with the X

axis .

(a) Find the angle between these rays and a roof R 0 ( ,

(b) Find the angle between these rays and the hori

z o ntal and n n n fro t pla es of projectio . A ray of light passes through the poin t p — 1)

and is reﬂected from a poin t r in a plane

s n b 0 mirror repre e ted y M ( , Find the angle * reﬂection and n o f n Of , the dista ce the poi t p from the

mirror .

Find the angle between a line with both projections parallel

to and n 0 the X axis the pla e Q ( , u est on n n S gg i . Si ce the pla e Of the given line and the

n ‘ n n perpe dicular must i clude a li e parallel to the X axis ,

its traces are parallel to the axis .

I n Fi 25 n L M N and L . N . 0 43 g two pla es , , are shown in a n and n n pictorial drawi g , we wish to determi e the a gle between

an n n n them . If from y poi t p we draw the li es p g a d p r perpen dicul ar n L N and L N 0 er en respectively to the pla es M , these p p dicul ar lines will determin e a plan e whi ch is perpendicular to both o f n n and n in n e and e r the give pla es , i tersects them the li es g . The

n e r n b n n n a gle q is therefore the a gle etwee the two give pla es .

n n o t n This a gle is usually determi ed directly . The method o r dinarily used is to ﬁnd the true angle g p r between the intersec t

' n nd r Art n ing li es p g a p ( . This a gle is the supplemen t of

n e r o f n e r r the a gle g . Of course either the a gles g (gt) o q p r ° n n n ( 180 is the a gle betwee the two pla es .

The ab o ve m eth o d i s p ractical ly th e sam e as ﬁn di ng th e angl e be tw een th e l in es cu t f ro m the two given pl an es b y an auxiliary pl an e passed thro ugh h m r n i r h i r l in e f i n rsec io n t e e e d cul a to t e O te t . , p p

T he angl e o f reﬂecti o n is equa l to th e angl e b etw ee n the ray an d the

irro r efo re reﬂectio n which i s call ed th e an l e o f incidence . m b , g

M 14 To nd the an le between two obli u e lanes. 44 . PROBLE . ﬁ g q p

ethod F an n in n n to M . rom y poi t space draw a li e perpe dicular

n n b n tw o i n each o f the given pla es . The true a gle etwee these l es

n ne is the angle betwee the pla s .

E XERCISE S

b n n S — 3 and Find the angle etwee the pla es , T 0 ( , — The corners Of a hip - roof are given by the poin ts a 2 §

" _ _ — ' _ _ ' b ’ 1 C 2 and d i : ( Fr i i ) y i i 1

— 2 n e — 2 , The ridge is the li e ,

—1 1 F n n b n es g , ( ) i d the a gle etwee the sid

c d an d a b e Use the of the Of the roof eg g . ( side traces planes ) (2) Find the an gle between the sides o f the

e a c roof a b g nd gb .

F n n eb en the n R — 1 an 80. i d the a gl etwe pla e , d a plane parallel to the X axis with its horizontal and front

traces respectively 1 inch behind and 1}1 in ches below

the X axis .

e ti on in n of o n o f th r Sugg s . The tersectio e e pe pen diculars with either the horizontal or the front plane

n l b n n is fou d easi y y usi g the side projectio s .

M n i o n 76 n t ake a drawi g sim lar to the ﬁgure p . , taki g he "h ° following data : Draw in n with a slope o f 30 to the ° = 60 a = 4 b = 2 c = 1 d = 2 e = . l X axis Take ﬁ , , , % , , i , 2 15 1 h a i n b in n f 1, g , Complete the dr w g y determ i g in o t and 0 accurately the l es q. ( 1) Find the angle between the roof plane t o p and the o ne

n r s in which the li e lies .

n r s n in F n (2) The li e represe ts a rafter the roof . i d the

an of b to be n gle the evels cut at its e ds , so that it may

e n r and to the o b joi ed to the ridge at , valley rafter g

s at . The corners and ridge of the irregular hip- roof shown In the

o f 7 b ﬁgure at the top p . 8 are located y the following

n : a 0 0 b 0 0 - 5 poi ts ( , , , , — d 0 - 4 l e — 3 D ( , , % , , eter mine the poin t e by ﬁnding the line Of intersection of the

n in e sides meeti g the ridge (g ) .

(a) Fin d the angle between the sides of the roof meeti ng

in the ridge . (b) Find the angle of bevel for the top side Of the hip

e c t ﬁnd an b rafter ; tha is , the gle etween the planes

d e c nd b c e of the sides a g.

c n m n n in b 0 e ( ) The li e represe ts a rafter the side g. Find the angle of bevel for its end at m to join the hip

e c . n n m n be rafter The fro t projectio of must supplied .

The b b - n r 83 ﬁgure elow shows a square utt joi ted hOppe . The edges of the outside top and bottom squares are re

ectivel -5 n and 2 n m n sp y i ches i ches . The outside pi g

n n F n edge is 3 i ches lo g . i d the angle o f bevel for the

n b n o f joi ts etwee the sides the hopper .

B the n in n s y ge eral method used the precedi g exercise .

ﬁnd n b n n 0 and the a gle etwee the pla e P ( ,

n n o f n the horizo tal pla e projectio .

° The horizon tal trac e of a plan e makes an angle of 45 with ° and n an n o f 60 the X axis , the pla e itself makes a gle

ront n . n n with the f pla e Draw the fro t trace of the pla e .

n for b in 2 an Take the data give the skew ridge Ex . 7 d

ﬁnd

(a) The angle between the plane Of the portal and a horizon

n tal pla e . (b) The angle between the plan e of the portal and a vertical

n n O f n d pla e through o e the e posts . (0) The angles between the p o rtal strut b d and the end posts d a b and c .

(d) The angle between the portal strut b d and the lin e Of intersection o f a vertical plane through o ne of the end posts with a plan e through the portal strut perpendicular f to the plane o the portal .

u esti on A S gg . simple way to pass a plane through a given line perpendicular to a given plane is to draw through any point in the given line a lin e that will be per

endicul ar n n p to the give pla e . The required plane is

n b tw o n n n determi ed y these i tersecti g li es . (e) Measurein degrees the angle found in (d) and the angles and i in B n marked a [ the ﬁgure . y trigo ometry the tan

n O f n n in d sin tan ge t the a gle fou d ( ) is equal to a ﬂ.

n n With this formula check the a gle fou d .

A i n n n n 4 5 . very s mple method for ﬁ di g the a gle betwee tw o

i in A r s n t. n sides o f an Object explai ed 38 . It is show there h ow

n n O f d c in Fi 22 an the a gle a betwee the side the roof i g. d the

n b n n pla e of the ase is fou d immediately from the projectio s . The

n b n d e and ron t n nn a gle etwee the side f the f pla e ca ot , however , n so n n the of n be fou d easily without ﬁ di g traces the pla es . The method to be explained in this article applies particularly to ﬁnding the angles between a plane given by its traces and the

n n horizontal and fro t pla es . Observe that the problem is the same

in 4 b w s . 8 . o n as that solved Ex The pro lem , ho ever , appears Ofte f for solution that a Special case should be made o it . If an oblique plane is pependicul ar to either the horizontal o r

ﬁ n n n o ne o f traces will be n the fro t pla e Of projectio , its perpe dicular

and b 10 n to the X axis , the other trace will show y its s pe the a gle

f n n i n o ne n o . I F the pla e makes with of the pla es projectio g.

' 26a a n n n n n and pla e , P , is show perpe dicular to the horizo tal pla e ,

the s10pe o f the horizontal trace shows the angle between the plane

n P and the front plane . The same pla e is shown by orthographic

i 26b and n e 0 n in F . r projectio g , the a gle mark d measures accu ately

the same angle . The problem is a little more complicated when the plane is

n of n in i 2 a I b n F . n oblique to oth pla es projectio as show g 7 .

of n b n one o f and this case , course , the a gle etwee the traces the X axis does no t show the angle between the plane P and a plane

n i n o f e n . a b proj ctio If , however , aux liary pla e , marked y its

A and F A so nt n trac es, H , is placed that it i ersects the pla e P, and is n i l a the n n as n in Fi perpe d cu r to horizo tal pla e show g.

2 7a the n 0 b n n n n of n , a gle etwee the li e of i tersectio the pla es P and A and the trace H A is the angle between the plane P and

o f n n the hori zontal plane . The method ﬁ di g this angle from the

ro ecti o ns is n in Fi 2 hi c . b orthograp p j show g 7 . The given plane

n and F and is show by its traces , H P P ; the auxiliary plane

A n to and F A n by its traces , H (perpe dicular H P) (perpe dicular

n n n o f to the X axis) . The li e of i tersectio these tw o planes is "h n and f in shown by its projectio s i j 17 j . This l e when revolved

n n al n b n n i h " i to the horizo t pla e a out its horizo tal projectio , j , as h ’ an a i n i and n 0 in its x s , takes the positio j shows the a gle true size . A di mension lin e marked s is shown to make the construction

ul be i n n . n A can plai er It sho d ev de t that the auxiliary pla e , , be drawn through any poin t in the trace o f the given plane

can ﬁnd n b n n Similarly we the a gle etwee the same pla e , P, and n n b n n F B O f an auxﬂ the fro t pla e y drawi g the fro t trace , i ary n B n n and pla e , , perpe dicular to the fro t trace Of P , making the

n B n to horizo tal trace , H , perpe dicular the X axis . The required

n n and n angle betwee the pla e , P, the fro t plane o f projection is shown by the true size of the angle between the front plane and

n of n n O f l n B n the li e i tersectio the auxi iary pla e , , with the pla e P .

s of n 4 6. The rever e this last operatio is made use o f in drawing the o f n n nl one of n on w n traces pla es whe o y them is give the dra i g .

If the horizontal trace of a plane is given and the angle the plane ma n n n kes with the horizo tal pla e is k own, the front trace can

n n in Fi 2 n e A ow . b b co structed . ssume that g 7 the horizo tal

n and n 0 n trace of the pla e P the a gle , , the pla e makes with the

n n n horizontal pla e are give . Through a y point in H P draw the

” h ’ n A o f an n and i horizo tal trace , H , auxiliary pla e , the line j making with H A an angle equal to the angle the plane makes with

n n a n n in the horizo tal pla e . This l st li e is the revolved positio the horizontal plane of a lin e i j of which we know the horizontal h ° h n i and n 3 of n b projectio , , the dista ce ( ) the poi t j elow the hori

ntal n n n o f b n Off on z o pla e . The fro t projectio j is located y layi g a h n n n s li e perpe dicular to the X axis through j the dista ce marked . The line i j determined now by two projection s is a line in the f r n and n in n . e equired pla e , j is a poi t its fro t trace The requir d front trace is drawn then through the proj ection jf and the in ter section o f the given horizon tal trace with the X axis . The method is exactly similar when the front trace and the angle

n s n n n and n al the pla e make with the fro t pla e are give , the horizo t trace is to be constru cted .

To nd the an le m de b a i ven lane wi th R M 15 . 4 7 . P OBLE ﬁ g a y g p o ei ther the hori z on tal or fron t planes of projecti n .

r n n n n ne Method . To measu e the i cli atio to the horizo tal pla Through any point in the horizontal trace of the grven plane pass be an auxiliary plane perpendicular to this trace . (It will

n b n perpendicular to the horizontal plane . ) The true a gle etwee the horiz ontal plan e and the lin e of intersection Of the auxiliary

n plane with the given plane is the required a gle .

T O meas ure the inclin ation to the fron t plan e : Pass an auxil n b n iary plane perpen dicular to the fron t trace . The true a gle etwee the front plane and the lin e Of intersection Of the auxiliary plane

n with the given plan e i s the required a gle .

EXERCISE S

0 120° and the F n n b n n ( , 87 . i d the a gles etwee the pla e P n horizon tal and fron t plan es Of projectio . 2 7 and n fo r b in Ex. 88 . Take the data give the skew ridge

ﬁnd the angle between the plane of the portal and the

n and n n of n horizo tal fro t pla es projectio .

8 M 1 Gi ven one trace and the an le a e la e R 6. v 4 . P OBLE g gi n p n

kes wi th the corres ondi n lane o ro ecti on to nd the o her ra e ma t t c . p g p f p j , ﬁ

ethod R e o f n M . evers the method the precedi g problem .

EXE RCISE S

° n n an n o f 315 The fro t trace Of a pla e , Q , makes a gle with ° n n n the X axis . The pla e itself makes a a gle of 45

n n D n with the fro t pla e . raw the horizo tal trace . ° n e o f n R an n 3 The horizo tal trac a pla e , , makes a gle of 0 ° n an n 45 with the X axis . The pla e itself makes a gle O f

n n . F with the horizo tal pla e Draw R. — — n o f - n b n a 4 2 The pla a dry dock is show y the poi ts , ,

b —2 — 4 c —1 and d — 3 T , , ( , , he planes o f the sides through a b and c d slope toward the ° and n 45 middle Of the dock , make a gles of with the

z n D o f n hori o tal . raw the traces the pla es of the two

n n sides me tio ed .

n n n n n and n 49 . Whe o e of the traces Of a pla e are give we k ow

n n n o f n onl y the a gles the pla e makes with the pla es projectio ,

b n I n . Fi 2 we have a more complicated pro lem tha the last g. 8a

n in n A r n . e re a pla e , P, is show a pictorial drawi g sphere is also p

n o n e an d n n n sented with its ce ter the X axis at , is ta ge t to the pla e

An n A is n n er endic i . u P , at auxiliary pla e , , represe ted as draw p p

n n and n in i and the lar to the fro t pla e , passi g through the po t

f e. n center o the sphere , This auxiliary pla e cuts a great circle

and n a b n n from the Sphere a li e from the pla e P . This last li e is

v n n or b n tangent to the circle at i . It is e ide t that the a gle etwee the line a b and the trace F A shows the angle between the plane P and

n n b b n n . a the fro t pla e If , the , the li e is revolved a out the trace ’ n n n l be n b a b F A into the fro t pla e , its revolved positio wi l show y . Lik ewise if the circular section cut from the sphere bythe auxiliary

n n n i be i plane is also revolved i to the fro t pla e it w ll a c rcle ,

n n w n f ~ o . B coi cidi g , however , ith the outli e the sphere y this revo l uti on n n n n a b and i to the fro t pla e , the li e the circular section O f

n in n an the sphere are show their true relative positio , d the point of

’ ’ n n i n i . n b n a b and F A ta ge cy , , is show at The a gle etwee , ’ f is O a . marked a , the true size

an n B is n in Similarly auxiliary pla e , , represe ted the ﬁgure , perpendicular to the horizontal plane and passing through the poin ts

n n n e and i . This pla e cuts from the give pla e , P , the lin e 0 d and

n n a circular sectio from the sphere . The a gle ﬂ between the line c d and the trace H B shows the angle between the plane P and the

n n B n c d and horizo tal pla e . y revolvi g the circular section in to

n n can be n the horizo tal pla e the true size of B show .

This method of analysis is most useful when the conditions — n nl n a and n and are reversed whe o y the a gles ﬁ are give the traces ,

and F be n . H P P , are to determi ed The actual process for the so i F 2 n i . l utio n of this case is shown g 8b. Through a point e on the

n . an n n n X axis as a ce ter draw a circle of y co ve ie t radius . This represents in orthographic projection the revolved position in either the horizontal or the front planes of any sections of the sphere

n s n n I n cut by auxiliary pla e passi g through the ce ter . order to

revolved osi ti on o the li ne o i ntersecti on n show the p f f with the pla e , P ,

f an n n nt n o auxiliary pla e perpe dicular to the fro pla e , draw the lin e

’ n n to an n n n n and a b ta ge t the circle at y co ve ie t poi t , draw a radius

’ e a so n b a e n n or b that the a gle is equal to the give a gle , , etween

’ n n e b the plane P and the fro t plane . The li e is drawn perpendicu ’ n a b e n lar to a e . The tria gle that is thus formed correspo ds to the

revolved position Of the triangle a b e shown very much fo reshort

’ w n e b Fi 2 2 a . no . b ened in Fig. 8 If the le gth ( g 8 ) is measured

nd O ff n b n n e a laid perpe dicular to the X axis ehi d the poi t , the h And n point marked b is Obtained . this is a poi t in the hori

n al . n n n n z o t trace H P Co ti ui g with the same ge eral method , draw

’ n n n n n and b c d tange t to the circle at a co ve ie t poi t , y drawing the

’ ’ 0 e o ff n . n n c d e d radius lay the a gle 5 The i tersectio Of with , ’ n e c n n d perpe dicular to , determi es the poi t , which is the revolved

n o n F L position of a poi t located the trace P . aying O ff then e d"

’ n df b n n equal to e d the poi t is O tai ed . If ow a e happened to be taken in such a position that it is perpendicul ar to F P in its

n be n d} true locatio , this trace could draw immediately through " and a and l be n b nd 0. ; , similarly , H P cou d draw through a

in a e n in an n n n n it S ce , however , was draw y co ve ie t directio ,

n o t be n n n to F but O bvi could take ecessarily perpe dicular P , , o usl an a e n n in y , if arc with a radius is draw as show the ﬁgure ,

’ n n n are and n F P must be draw ta ge t to this through the poi t d . For the same reason H P will be drawn tangent to the arc with h a radius c e and through the point b

M 1 To nd the traces o 0 R . a la e 5 . P OBLE 7 ﬁ f p n when the angles the lane makes wi th both the hori z ontal and ront lanes are ve p f p gi n .

e hod n n M t . Imagine a sphere placed ta ge t to the plane Of which

be n and tw o n ‘ the traces are to fou d , auxiliary pla es are passed through the center Of the sphere and through the point where the — s e n o ne n n n pher touches the pla e perpe dicular to the fro t pla e ,

n u to n n the other perpe dic lar the horizo tal pla e . These planes will each cut a circle from the sphere and a tangent line from the

n n n n n give pla e . The a gles these ta ge ts make with the fron t and

n n e n n n horizo tal pla es , resp ctively , are the a gles the give pla e makes

n n n with the front a d horizo tal pla es .

E XERCISE S

° o f n n f 45 Draw the traces a pla e , P , which makes a gles o ° and 60 n and n n respectively with the horizo tal fro t pla es . ° Draw the traces of a plane which is in clin ed 120 to the ° z n n and 5 to n n hori o tal pla e 7 the fro t pla e . ° The side of a bridge pier in a river makes an angle o f 60 ° n o f and 50 n with the pla e the water , with a vertical pla e

n of R at right a gles to the course the river . epresent the plane o f the side of the pier by its traces on assumed

n planes o f projectio .

To nd the tru e di s r m a M 18 . tance o i ven o nt 5 1 . PROBLE ﬁ f g p i

ne to a gi ven li .

D n lane n and n Method . etermi e the p Of the poi t the li e by

drawing through the point a line parallel to or intersecting the

n n F n n n b n w give li e . i d the i tersectio s of oth li es ith either the

n n n n horizo tal or the fro t pla e . With a trace of this pla e as an

n n n n n o f n axis , revolve these li es i to the correspo di g pla e projectio , n n a d measure the dista ce required .

E XERCISE S — F n s n n i 2 — l - L to i d the di ta ce from the poi t , , }) the

° n — 2 — — k — ~ — l — li e 7 , t, i ) , i . i , t) The telephone wires running from a village to a house o n the side o f a mountain follow the shortest lin e between

n v - 2 and h — 3 0 A the poi ts ) , , camp at c is to be connected to the telephone system of the village by erecting a line join ing the one

n h n between v a d . Draw the projectio s of the shortest lin e that' can be put up ; and ﬁnd the length of the line

thus located .

n n n in 62 ﬁnd n I the tria gle give Ex . the dista ce from

the poin t e to the line a b. For supporting a crane a cable is to be attached at o ne

end n c — 1 — 1 — 2 o n and to a poi t , , ) its mast , at the other end to a steel beam a b

— n r c 1) o n an adjoining buildi g . Show the p oje

f can be u tions o the shortest cable that sed .

To nd the shortest di stance between two l nes M 19 . i 52 . PROBLE ﬁ

not i n the same plane.

ho w o n a b and c d n o t in n wn Met d . T li es the same pla e are sho

a b d Ar i 2 n N c t. 9 . in F g. Pass a pla e , M , through parallel to ( ’ n At n e Project 0 d upon this pla e . the poi t where the pro

’ ’ ction c d . n a b n n ne je i tersects , draw a li e perpe dicular to the pla ' e n t o f t e n e e intersecting c d at . The true le g h h perpe dicular

r i n e is the equ red dista c .

E XE RCISES

F n t s n b n the n s to l and m n i 99 . i d the shor e t dista ce etwee li e n

25 . Ex . Find the distance between a line from a to b and the mast

in 7 1 . o f the derrick Ex . Through each o f tw o lines pass a plane parallel to the other

F n t i n b n tw o line . i d the shor est d sta ce etwee these

planes . f 65 b o . Solve Ex . y the method this article

R B M 2 0 Given the ro ecti ons o the center o a ci rcle 53 . P O LE . p j f f o kn own di ameter to draw i ts ro ecti ons so that i t shall li e i n a f , p j gi ven plane. h e hod n f n n Mt . The ce ter o a circle is show by its projectio s c

d i i n in n n n n F . 30 and g as lyi g the give pla e P . The horizo tal and front projections o f the circle are to be drawn so that the

n n o f circle also shall lie in the pla e . The ce ter the circle is revolved ' b F P as an axis into n n and o f a out the fro t pla e , the true size the

n n i b n n is . circle draw Whe th s circle is revolved ack i to the pla e P ,

n ll be b n o f o n both projectio s wi ellipses , ecause the projectio s a circle

n n b n pla es of projectio that are O lique to its pla e are ellipses . The length o f the major axis of the ellipse in the front projection is the

o f and o f F same as the diameter the circle is , course , parallel to P . As all lengths which are perpendicular to the axis of revolution

r h n in n n n is a e fores orte ed the fro t projectio , the mi or axis the fore

n F ho rtened diameter perpe dicular to P . The front projections

n n o n n o f n n n n o f the mi or axis is , the , a li e i deﬁ ite le gth draw through

n c perpendicular to F P . Mark the poi t where this projection

n if and n i 0 n n crosses the fro t trace , revolve the li e i to the fro t plane f f ’ b i cf as an . r n 1 n b i c a out axis The revolved posit o s show y ,

’ n c n F m on which true le gths a be measured . ro c lay off the length ’ ’ c e r I n n - n equal to the radius ( ) of the circle . cou ter revolutio ’ ’ n 0 b c} and e ef n erof the poi t revolves ack to to . The is the semi i n of l n n n m or axis the el iptical fro t projectio of the give circle .

The horizontal projection of the circle is then easily found by

n h as drawing lines that lie in the pla e , P (suc diameters through

c and n n n n) and ro ecti n the center , , ta ge ts to the fro t projectio p j g

points on them from the fron t projection o f the circle . The hori

z ontal projection o f the circle could be found also by repeating the

n n method used for obtain ing the fro t projectio .

b an n n in n n 54 . This pro lem has importa t applicatio determi i g the plane Of guide pulleys to direct belts runn ing between pulleys ? I i n F . 3 which are on shafts at right angles to each other . g 1 two

pulleys are shown with centers at e and b. The direction of motion

is n b be O ff b in show by arrows . The elt must led the pulley at

l ain ts n i ts n and on i . pla e , led the pul ey at pla e To accomplish this , h a guide pulley is needed to direct the belt . Usually t e guide pulley may be placed at any convenient point between the two pulleys

n n n at a a d at b. Select a poi t where the directio Of the belt is to

“ " n n d in and lin s d be e . an cha ged , as the poi t the ﬁgure , draw the f d

n d e tangent to the pulleys . The pla e of the guide pulley must

n in n o f tw o n n n n be the the pla e these ta ge t li es . This pla e

and F n is shown by the traces H P P . The projectio s o f the guide pul ley are found by revolving the lines f d and d e into the n n n b an horizo tal pla e of projectio a out H P as axis . The revolved

’ position o f d is shown at d The actual size O f the guide pulley is

n n n n f d an d e r show ta ge t to the revolved positio s O f d . A fter evolv ’ in b n n center c O f g ack i to the pla e P , the ( ) the guide pulley is shown h n c n cf n O f by its projectio s a d . The projectio s the guide pu lley

n b n in are fou d y the method explai ed the last article . The shaft Of the guide pulley being perpendicular to its plane is shown by

n n corres o nnin projectio s perpe dicular to the p g traces .

i 31 o n o ne b O ff I n F . a in l n e g , side , the elt is led the pulley i with

o f ul b so on is no the rim the p ley , that this side there change in

f I n n o b . n the directio the elt practice , however , very ofte the given pulleys are not placed so advantageously and two guide pul leys are needed .

EXERCISE S

103. n o f 1 n in Draw the projectio s a circle , diameter 5 i ches , ly g in n —3 the pla e Q , 104 n on n n . Two pulleys revolvi g li es of shafti g at right angles

be nn to each other are to co ected by a belt . Determine the planes of intermediate pulleys to properly direct

b n n the elt , drawi g also the projectio s of all the pulleys

n that are eeded .

2 1 ven a s r e b s o . R M . Gi u ac i t c ntou red lan and 55 P OBLE f y p , a lane b i ts slo e to determi ne the secti on o the sur ace cu t b he p y p , f f y t

n n f Method . Draw the horizo tal projectio s o a number o f

n n n in n n and n horizo tal li es lyi g the give pla e , havi g the same

n indices as the contour lines . The poi ts where these lines meet the contour lines with the same indices are points in the required

e n b n b w n section . The compl te sectio is O tai ed y dra i g a fair curve thr ough the poin ts thus Obtained . E XERCISES b n b n n 5 . 10 . The ﬁgure elow represe ts a hill y its co tour li es Draw plan es to show the embankments o f a railroad “ ” n in n f cut passi g through it . Show the tersectio s o

these planes with the surface of the hill .

l n Contou r l i nes are u sed to j o in po in ts a t th e sam e e evatio above a pl ane r Numbers o n th ese l ines c ll u m e to b e z e o . a e o f which th e el evati o n i s ass d , d l e ati o ns dices re resen t the e . i n , p v

CHAPTER III

PRO BLEMS RE LATING TO PLANES TANGE NT TO SOLI D S

6. u a be n b n n 5 Every s rface m y ge erated y the motio of a li e , and the different positions assumed by this line are called the

e e ents o f r l m the su face . A plane surface or a plane is generated by a straight line moving along an other straight line and remain ing always parallel to its ﬁrst

n * positio . A sin gl e curved surface IS generated by a straight line moving

so an two o f ns n in n that y its co ecutive positio s are the same pla e . A w arped o r twisted surface is generated by a straight line moving so that n o tw o o f its consecutive positions are in the same

n pla e .

n n u f and n Pla es , si gle c rved sur aces , warped surfaces are ge erated b n n and n y the motio of a straight li e , therefore all have eleme ts

in . that are straight l es Every warped surface , however , is curved , and it is therefore possible to conceive it also as being generated by a as it s n n l n n curve which , move , co ti ua ly cha ges its form accordi g

o n l t a deﬁ ite aw . A doubl e curved surface is generated by a curve moving along

n no n n . a other curve . It has eleme ts that are straight li es

A c ne n n b t 57 . o is a si gle curved surface ge erated y a straigh line moving along a curve and also passing through a point not in

tex n n f n er . the pla e o the curve . This poi t is the v Of the co e A cylin der is a single curved surface generated by a straight line

n l be moving along a cur ve with all its positio s paral el . It may

n n regarded as a Special case o f a cone with the vertex at i ﬁ ity .

he ro er i es o f l an es w as sum ed in th e e i nn in A kn o wl edge o f t p p t p as b g g. e meth o ds o f re resen tin l anes and o in ts an d lines in th em ha e een Th p g p , p , v b n h a te discussed i n the precedi g c p rs .

Problems relating to cylinders are solved therefore by the same

b n n methods that are applied for similar pro lems relati g to co es .

A n n n n n o f pla e cutti g all the straight li e eleme ts of a co e , or

n in in bas a cyli der , tersects it a curve called the e . If all the elemen ts of a con e make the same angle with a straight

n n h n li e passi g through the vertex , it is a rig t co e ; otherwise , it is

n n n an Oblique co e . If all the eleme ts of a cyli der are perpendicular

b r h c l n de is an b e c lin to the ase , it is a ig t y i r ; otherwise , it o liqu y )

der .

8 A c n u is n n 5 . o vol te a Si gle curved surface ge erated by a straight

n movrn n b SO t li e g alo g a curve of dou le curvature , hat it is always

n n n n o f ta ge t to the curve . The co secutive positio s the straight

n n n n and n o n li e ge erati g the surface i tersect two two , three i ter

c in in n n se t g a commo poi t . There are as many kinds o f convolutes as there are curves of

m n a li double curvature . So e are importa t for their practical pp

n o f n in and n n catio s . The methods co struct g represe ti g them will

i Ar e n t 76. b discussed .

A s r a e o f r l n ne n rev 59 . u f c evo utio is o that is ge erated by the

o l u ti o n n n b n in of a straight li e , or a pla e curve a out a straight li e the same plane as an axis .

n n n — There are o ly two Si gle curved surfaces of revolutio , the

n and n n b right co e the right cyli der , whe they have circular ases .

n b O f n The pri cipal dou le curved surfaces revolutio are the Sphere ,

b and b — the ellipsoid , the torus , the para oloid , the hyper oloid sur faces which are explained later .

6 A n in an n n b 0 . poi t y pla e or curved surface is determi ed y

n su r ace n b n tw o n . projectio s If , the , a f is give y the projectio s of

n and oi nt on n b n o ne n its outli es , a p the surface is give y o ly projectio ,

n n can be n in n a other projectio located , after determi g two projectio s

s o he t of an element of the surface pas i ng thr u gh t poi n .

2 con e n b z n and n n I n Fig. 3 a is Show y the hori o tal fro t projectio s

A n a on ~ n n b Of its outlines . poi t the surface Of the co e is give y its

n horizontal projectio . The horizontal projection o f an element h f n n a an As n a o the co e is draw through d the vertex . the poi t

be on of n n n n o r may the top side the co e earest the horizo tal pla e , o n n b v and c v z n the lower side , there are two eleme ts , , with hori o tal h n n a projectio s passi g through . The front projections of these

n bf vi and cf vf n and two eleme ts , , are draw through the vertex through the front projections of the intersection s of the two ele

n t b n n O f n me ts wi h the ase, The required fro t projectio s the poi t a n o n n bf vi and of vi a f and a f are the the eleme ts , at l z . For the case Of the cyli nder the same method o f solution is applicable .

6 1 . M 22 Gi ven one ro ecti on o a o nt on the sur PROBLE . p j f p i

ace o a cone to nd the other ro ecti on f f , ﬁ p j .

ho n n f n Met d . Through the give projectio O the point a d the

c n ‘ raw n F n vertex of the o e d eleme ts Of the surface . i d the required

n f n n n f n projectio s o the poi t o the other projectio s o these eleme ts .

62 R M 23 Gi ven one ro ecti on o a o nt on the sur . P OBLE . p j f p i

e o li nder to nd the other ro ect on ac a c i . f f y , ﬁ p j

e hod n n f M t . Through the give projectio o the point draw

n o f to an n A s in the eleme ts the surface parallel y eleme t shown .

of n n n o f nt case the co e precedi g , the required projectio s the poi are n o n n o f n fou d the other projectio s these eleme ts .

6 n a in on b 3 . Whe po t a dou le curved surface , such as a Sphere

n b n o ne n f n . t ellipsoid , torus , etc , is give y o ly projectio , a di fere method is used for ﬁnding the other projection o f the poin t o n the

n n and n I n Fi surface tha that employed for the co e the cyli der . g.

n and n n O f an 33 the horizo tal fro t projectio s ellipsoid are shown .

A n a on of n b n poi t , , the surface the ellipsoid is give y its fro t pro

i i A n n a and n f ect on a . o j pla e , P , is draw through the ce ter the

n n n s ellipsoid perpendicular to the fro t pla e . Its fro t trace i

n n SO marked F P . If this pla e is the revolved that it is parallel

’ ’ n n nt n b F and to the horizo tal pla e , the fro trace is show y P , the ’ ’ n F elli front projection of a is at M. The pla e P cuts from the p s id secti on o f the ~h ori z ontal n o f a o a , which projectio the surface is n

exact representation ; and a (the revolved position of a) has its h I n n n a . n n horizo tal projectio at revolvi g the pla e P, back to its

orr l nal n n a a in g positio , the poi t moves to a circular arc lyi ng m

n n n a vertical pla e . The fro t projectio of this are is M of ; an d its h h n n O f a a t horizo tal projectio is , course , , parallel o the X axis . h z n n a n i n b The required hori o tal projectio , , is the determ ed y n n n af b drawi g projecti g li es from . O serve that the solution gives

n n a h h f o n o . also a other projecti , l , s ow at the top the ﬁgure

6 24 e R . v n o e 4 . P OBLEM Gi n projecti on of a poi nt on a dou ble curved sur ace to nd the other ro ecti on f , ﬁ p j .

ethod n M . Through the give projection of the point and the center of the double curved surface draw the trace of a plane perpendicular to the plane of projection in which the projection o f

n n R the poi t is give . evolve this plane SO that it becomes parallel

n n O f rO i n D to a seco d pla e p ject o . etermine the projections of . n in n B the poi t its revolved positio . y revolving back to the

n n n o f n * origi al positio , the required projectio the poi t is located .

6 . I n n n n n t r 5 ge eral , a pla e is ta ge to a su face at a given point

n n n tw o n n whe it passes through ta ge ts to li es of the surface , meeti g in n n . n n n t an n the give poi t If , the , through a give poi , y two i ter sectin n n and n n in g li es of the surface are draw , a ta ge t to each l e is

n n an is n b n n draw at the poi t , the required pl e determi ed y the ta ge ts .

Fo n a n n n n r drawi g pla e ta ge t to a si gle curved surface , the

n I n e t . con ge eral method is somewha simpliﬁed the case of the , we may observe that if a plan e is tangen t to the surface at a given

n n n t n an poi t , it is ta ge t to the surface thoughou the le gth of

An o ther meth o d i s to draw a right cyl i n der w ith its axi s o n th e axi s o f th e do u l e cur ed su rface and w ith o n e o f its ases assin th ro u h th e ro b v , b p g g p n f h e o in th i i en I n Fi 34 th e o int a i s i en th e ro ecti o O t t at s . j p g v . g p g v by p

f in h e r ce in u re D raw th ro u h (Lf th e f ro nt ro ec tio n jectio n o as t p e d g ﬁg . g p j

o f the ase O f a ri ht c li n der The h o ri z o n tal ro ec ti o ns o f the ase are b g y . p j b h h h h b c an d b an d O f th o in t a are a an d a , , , e p , h e its h o ri z o n tal ro ec ti o n as d I f th e pro j ecti o n o f a po int 1s gi v n by p j , f h its fro nt pro j ecti o n i s in an arc wi th th e fro n t pro j ecti o n o f the axi s O t e h u rface as a cen ter an a r iu s e ual to th e h o ri z o n tal di stan ce f ro m e to s , d ad q f h e o u l in h elli so i Th e f ront ro ecti o n s are th en d and d . t t e o f t e p d . p j l "

n n n n and is n nt eleme t passi g through the give poi t , therefore ta ge

n n Th to the cone at the poi t where this eleme t meets the base . e construction o f tange nts at the gi ven poi nt makes it necessary to n n of n b s represe t sectio s the co e , which is usually a la oriou

A n n to he base n n n process . ta ge t t at its i tersectio with the eleme t is l n and n of n n a more easi y co structed , is used i stead the ta ge t to

i n section at the g ven poi t . i i 5 n n n . I n F . 3 n a o g a co e s Show with a poi t , , marked its surface The plane tangent to the cone at this point is determined by the element b v through a and by a tangent b c to the curve o f the base

n n v w o n n n b v and b c at its i tersectio with b . T i tersecti g li es

n n n tan are thus represe ted . They determi e the pla e , P , which is

n n gent to the co e at the poi t a . As a cylinder may be regarded as a special case o f a cone with the n n b n n ll vertex at i ﬁ ity , the pro lems relati g to the cyli der wi be solved by the same methods as for similar problems relating

n to the co e .

66 R 2 To ass a ne li n M 5 . la ta o a cone or . P OBLE p p ngent t a cy der thro u h a i ven oi n t on the su r ace g g p f .

o n n f Meth d . Through the give poi t draw an element O the

A n n a . t nt b surface the i tersectio Of this eleme with the ase , draw

n n b line tangent to the base . The required pla e is determi ed y the

n n n n in eleme t a d this ta ge t l e .

E XERCISE S

a n n n an b e n a 106. P ss a pla e ta ge t to O liqu cyli der through point b o n its surface :

1 b o f an n i n in r 07 . The ase i verted right c rcular co e is the ho i

z ontal n and n b n n pla e , the a gle etwee the eleme ts of the surfac e and the axis is Pass a plane tangent to

n n the co e through a point c o its surface .

A n be wn n n cone a l so 67 . pla e may dra ta ge t to a through a

u tside B f n point which is o the surface . oth the vertex o the cone a d

n n b of in n n n A n the give poi t must e course the ta ge t pla e . li e

n n and n n n nt joining the vertex and the give poi t , a other li e ta ge

nd n n n n to the base o f the cone a i tersecti g the ﬁrst li e , determi e the tangent plane . A 36 cone and nt 0 of n . I n Fig. a a poi outside it are show

n 0 nd v line 0 v is drawn through the poi t a the vertex . Through the

n B of b n i t n intersection of o v with the pla e , , the ase , the li e is draw

n n n n b tangent to the base . The ta ge t pla e P is determi ed y these t lines o v and i . When for the same conditions a cyli nder is used instead of a

n n n n co e , the solutio is the same except that the li e draw through

n is b one n the vertex for the co e , replaced y through the give

n point parallel to an eleme t .

To ass a lane tan ent to a o e o l 8 M 26 . c n r a e i n 6 . PROBLE p p g y

ou tside the sur ace der through a gi ven poi nt f .

e ho n n n n M t d . Through the give poi t draw the projectio Of a li e

in the o f n the and which , case the co e , passes through vertex , , i f n to an n n o . the case the cyli der , is parallel eleme t Produce this line to intersect the plane o f the base and draw a tangen t to the

n n n base through the poi t o f i tersectio . The required plane is de termined by this tangent line and the lin e already drawn through the given point .

E XERCISES

Pass a plane tangent to an oblique cone and through a point

in the X axis .

' Pass a pl ane tangent to a cylinder which has one base

in n n and n in n n the horizo tal pla e a other the fro t pla e , and throu h n in n n b g a poi t the fro t pla e outside the ase .

6 . R 2 To ass a la e 9 P OBLEM 7 . p p n tangent to a cone and parallel to a ve e gi n li n .

etho M d . Through the vertex Of the cone draw a lin e parallel n At n to the given li e . the poi t where this line in tersects the plane

b n n nt Of the ase , draw a li e ta ge to the base . These two lines de

n n termi e the required pla e .

EXERCISE S

Pass a plane tangent to an inverted Oblique cone and paral

n n lel to a Oblique li e . Pass a plane tangent to a right circular cone and parallel

to a line in the side plane . Pass a plane tangent to a right circul ar cone and parallel

n n to a Oblique li e .

o ass t 0 R M 28 . T a lane an ent to a 7 . P OBLE p p g cyli nder and

en parallel to a gi v li ne.

e ho an n in n n M t d . Through y poi t the give li e draw a line parallel

n n to an element o f the cyli der . The pla e determined by the given

n and l n n be arallel n li e the i e just draw will p to the required pla e . A plane tangent to the cylinder is then determined by a line tan

n b and a to n n and ge t to the ase p rallel the pla e already fou d , the e n o f n an nt in n leme t the cyli der which the t ge l e i tersects .

E XE RCISE S

Pass a plane tangent to an Oblique cylinder and parallel

to the X axis . Pass a plane tangent to a right circular cylinder with its

ba in an b n and t n se o lique pla e , parallel o a li e in the

same Oblique plane .

M 2 To ass a lane ta ent to a s r 1 R 9 . n u ace o r v 7 . P OBLE p p g f f e o

o h a ven oi nt on its su r ace luti on thr ug gi p f .

n n n n Method . Draw through the give poi t a li e ta gent to the intersection o f the surface with a plane passing through the poin t

n n b . n and the axis . Whe this li e is revolved a out the axis it ge erates a right cone tangent to the surface in a circumference which con

n n A n n n to th n n tains the give poi t . pla e ta ge t e co e at the give

n n n nt to r a poi t is the required pla e ta ge the su f ce.

E XERCISE S

5 h an nt a o n s o f an wi t a 11 . T rough y poi , , the urface ellipsoid h

n n n vertical axis , pass a ta ge t pla e .

A 1 n in t n n 116. circle i ch diame er with its pla e perpe dicular

to n n revolves abou t a verti cal a xi s the horizo tal pla e ,

is n e n o f A which three i ch s from the ce ter the circle .

rfa of n a toru s n su ce revolutio called is thus ge erated .

h a nt b on t a n n n Throug poi his surf ce draw a ta ge t pla e .

To ass a lane tan ent to a s here an 2 LE 30. 7 . PRO B M p p g p d

v li through a gi en ne.

od an il ne r n to Meth . Pass aux iary pla pe pe dicular the given

n o f n line through the ce ter the sphere . (This pla e cuts a great circle from the sphere and will cut a line tangent to this circle from the

n R l n i requi red tangent pla e . ) evolve the auxi iary pla e with ts intersections with the given line and the sphere in to a plane o f pro

n n n o f n n jection . Through the poi t Of i tersectio the give li e with the auxiliary plane draw a lin e tangent to the circular intersection

n A n of the auxiliary pla e with the Sphere . pla e passed through th e tangent line (when revolved to its o riginal o r true position)

lin is one e e and the given e the r quir d .

EXERCISES

Draw a pl ane tangent to a Sphere and passing through

n b n a y o lique li e . Pass a plane through a line parallel to the X axis and tangent to a sphere with its center in the horizon tal

n pla e .

n i C nv l utes . I F . 37 a wi 73 . o o g regular prism th sixteen sides

The n now f n t of is Shown . stude t must orm a me tal pic ure a right

n o f n b r n on the h k tria gle made thi card oard , esti g proj ection i 3

n be l in the ﬁgure . This projectio shall a so the horizontal leg o f the triangle . The vertical leg is Shown in the front projection of

b n 17 to c the prism y the dista ce from the base . We an imagine then such a paper triangle wrapped around the prism with the ver tical leg through i held station ary and the long leg of the triangle

b n always touches the ase of the prism . The fro t projection of the hypothenuse of this paper triangle is then a broken line

nn n f nt n n i h c b a co ecti g the ro projectio s Of the poi ts , , g, , , . Each portion of this broken lin e is equ ally in clin ed to the edges

f n n o the prism . The horizo tal projectio Of the hypothenuse is

b n n in n n n n the roke li e jo i g the horizo tal projectio s of the same poi ts .

o f n is now n in The surface the paper tria gle u wrapped , tak g it off

n And it n n o e Side of the prism at a time . if is tur ed o each

in i n as on n n edge of the prism success o a hi ge , till each u wrapped

n in n n n f portio co cides with the pla e of the ext Side , the portio o the hypothen use that has been released becomes an extension and

n n n o f n n n a ta ge t Of the portio the broke li e o that side of the prism .

n n b n a and b n a b Thus whe the portio etwee is u wr pped , it ecomes tangent to the Side in which b c lies ; and when also the portion b n b and c s n n a b c o f n is etwee i u rolled , the portio the hypothe use

n n c n n n 0 d ta ge t to the side o tai i g . Let us now consider the properties Of this broken line from

in c b c n c n a b b d . t a to i . The portio tersects , i tersects , etc ; tha “ ” is n t tw o and t o but a b not in c d , they i tersec w ; does tersect

d e tc r b c n t e . no does i tersec ,

n n b n n on 74 . Co sider also a other property of the roke li e the

n b n n n prism . If the um er of Sides of the prism is i creased i deﬁ itely ,

h e ix the broken lin e o n its surface becomes a curve called a l . It is a curve generated by a poin t moving o n the surface Of a cylinder

O n n n n Of revolution S as to cut all the eleme ts at a co sta t a gle . The

n n ar un d and moving point has u iform motio o , at the same time

ara e o f n . n n p ll l , to the axis the cyli der The method for co structi g i i n in F 38 . S . O a helix is Show g The axis of the hel x is vertical , that the horizontal projection is a circle with its cen ter at o in

n n n i n the axis . Now if the ge erati g poi t moves the d sta ce from

i f in n one n b the m to v maki g complete revolutio a out axis ,

n m s t u and v n passing through the poi ts , p, , , , , the vertical dista ce f between W and v is called the pitch Of the helix .

A cu ed su rf a e f n h 75 . rv c is formed o the tange ts to t e

b n n in Fi 3 n n b O f O f roke li e g. 7 whe the um er Sides the prism b n n n n and b n . has ee i creased i deﬁ itely , it ecomes a cyli der

n of n m b n c o d e The eleme ts this surface are the li es , , , p , q f,

n n n n c i n de con etc . Such eleme ts whe ta ge t to a yl r form a

e su face n of n in volut r . Portio s such surfaces are represe ted

F nd 4 n b s a h 39 a 0. igs . They are surfaces ge erated y a tr ig t

n e m in a n ur e o f d b e cur a u e so h a th e n e li ov g lo g a c v ou l v t r , t t li i s a n n o th e cu r e I n n n a s a e t . a lw y t g t v this surface , agai , y two , but no n in n in three , co secutive straight l e eleme t lie the same

n pla e .

n in Fi 40 can be The co volute surface g. regarded as formed by the consecutive position s of the hypothenuse O f a paper triangle as it is unwound from the surface of a cylin der o f which the base h h h " a d n e n a end n n n is the circle . The poi t at the of the u wi di g h h h h n in a c i in z n n tria gle will always lie the curve f the hori o tal pla e . This curve is the invo lute of the circular base of the cylinder ; but z n o f n it is also the hori o tal trace the co volute surface . The

n is n u b cyli der itself o part of the s rface . O serve the striki ng resemblance between the way this surface winds around a cylinder n n f — and the co volutio s o a sea Shell . When a helical convolute is to be represented on a drawing the curve of the helix to which the surface is tangen t should — e n in and n b accurately co structed , draw g ﬁrst the top fro t views

n o n . Of the cyli der which it lies If the axis is vertical , with the

in n n n b base the horizo tal pla e , it is show a ove that the horizon

n n i tal trace of the co volute is the i volute Of the c rcular base .

i 40 6 n m o n n in F . 7 . If a poi t the co volute surface g is given h n b n n m n n can o ly y the horizo tal projectio , the fro t projectio be

n n n n o f an found by co structio . The horizo tal projectio element

I f a tan en t roll s u o n a ed cu r e an o in t o mi t descri es a sec g p ﬁx v , y p b o n d ll n in o l u e f he r I n Fi 40 su o se a th read to cu rve ca ed a v t O t ﬁ st . g. pp be w o un d o n the ci rcul ar b ase an d kept tau t as it is u n w o un d fro m the end O f the h h h " h read at a then the en d at a will descri e the cur e a c i call e t , b v f , d the

i nvolute o f th e circl e . passing through the point m can be drawn tangent to the circul ar " " n and n n nt b i . ase at , it will i tersect the i volute at The fro f d n o n an i in rd and 17. projectio s are respectively the helix at , at h B n n i n} i f and n n m y drawi g the projectio a projecti g li e from , the f fron t projection o m is determi ned . The solution can be reversed : If the front projection Of a point on n n n n can be the co volute surface is give , the horizo tal projectio

n fou d .

R 31 Gi ven one ro ecti on o a oi nt on a convolute 77 . P OBLEM . p j f p

ethod n o f an n n M . Draw a projectio eleme t through the give

n n D n n projec tio Of the poi t . etermi e the other proj ectio of this

n f element and locate on it the required projectio o the point .

E XE RCISE S

Draw the projections o f ten equidis tant elements o f a

n helical co volute surface with a vertical axis . The diameter and pitch of the helix are respectively 2 inches

n a d 3 inches . Find the intersections o f the convolute surface given in

1 1 h t o n r n ar Ex . 9 wit w pla es pe pe dic ul to the axis of

i of the hel x . Observe the Shape the c urves cut by these

n pla es .

Draw the projections o f six elements o f a convolute sur

f O f is to ace which the ax is parallel the X axis . The diameter and pitch of the helix are respectively2 % in ches

n 4 n A n a d i ches . ssume o e projection of a point o n the

and surface locate the other projection . (Use the Side

plane . )

D tw o one - n an raw helices , right ha ded d the other left

n on n e 17 n c I n i ha ded , a cyli d r } i hes d ameter and 2 inches

be 1 n an high , the former to Of i ch pitch d the latter

1 n Of 2 i ch pitch .

Represen t a square- threaded screw Of the following

n n : t o f is 2 7 n dime sio s Ou side diameter the thread } i ches.

b o f 1 n Diameter at the ottom the thread is 5 i ches .

1 n O f i n . s n Pitch is i ch Thick ess the thread 4 i ch .

n Show two complete tur s of the thread .

A n O f a Spiral spri g is the form of a square screw thre d .

- n n and The cross sectio is 4 i ch square , the outside diam

and 3 n and 2 n eter pitch are respectively i ches i ches .

n n Draw two complete tur s of the spri g . ” A Spiral Spring is made Of round wire as Shown in the ﬁgure

« b . o f n elow The diameter the wire is 5 i ch . The outside diameter O f the spring is 2 } inches and the pitch is 2

n th n and n i ches . Draw e pla elevatio of tw o turns Of

n the Spri g .

u esti on n S . n gg If a Spiral Spri g is made Of rou d wire, we conceive its surface to be generated by a sphere mov

i n n n n o f g alo g a helix which is the ce ter li e the wire . The projections o f the helix are ﬁrst drawn and then the projections of the sphere in a number o f different

n positio s .

8 2 To ass a lane tan en t to a convolu te sur R 3 . 7 . P OBLEM p p g

e hr u h a i ven oi nt on the sur ace fac t o g g p f .

hod n r n Met . (The same as for the co e o the cyli der . ) Through

n n At n n the given poi t draw an eleme t of the surface . the i tersectio

n n i n of this element with the base draw a ta ge t . The requ red pla e

n and n n n is determined by the eleme t the ta ge t li e .

To ass a lane tan ent to a convolute sur M 3 . ace 79 . PROBLE 3 p p g f

o s de the s r a e and through a gi ven poi nt ut i u f c .

ethod n n n hi M . Through the give poi t pass a p la e w ch is per

l r n n is n pendicu a to the axis of the Surface . The i tersectio of th pla e

an n to n n n n with the surface is i volute , which a ta ge t is the draw

n n n o f a n from the give point . Draw a eleme t the surf ce at the poi t

f n his e nt and n n n i n o tange cy . T leme the ta ge t li e determ e

n required pla e .

E XERCISES

2 n t n n on e on 1 6. Draw a pla e a ge t to the helical c volut c structed

in 1 19 at an n on . Ex . , y poi t the surface

n n n n a c n 127 . Draw a pla e ta ge t to the co volute surf ce o structed 2 n in 1 1 a d t an t o ts the . in Ex. hrough y po u ide surface

CHAPTER IV

INTE RSECTIONS AND D E VE LO PME NTS OF SOLID S

80 in ersec n of an su r ace h a i en an e un . The t tio y f wit g v pl is fo d

n o f auxi li ar an es I n cu by passi g a series y pl , such a way that they t

r stra ht lin es c rc es o r h er cur es th at can from the su face , ig , i l ot v be u ck d awn an n s r h lines d n a t . q i ly r ; from the give pla e , t ig The

nt n o f n l n i ersectio s these li es cut from the surface , with the i es cut

n n n on n n from the pla e , determi e poi ts the required i tersectio . When straight lin e elemen ts can be cut from the surface the aux iliary planes are usually passed perpendicular to one o f the planes o f n n can be projectio . Whe circles cut from the surface , the aux iliary planes should be drawn parallel to the plane o f projection on

n of w l in which the projectio s the circles i l appear their true form . The tan gent to a curve o f intersection at a given point on a sur face lies in the plane which cuts from the surface this line o f in

i s n in n n n a tersect on . It is al o a li e the ta ge t pla e to the surface t

n n the give poi t .

8 1 o f n n o f an b ue c n e n . The curve i tersectio o liq o with a pla e P

in Fi 4 1 v o f n n n is Shown g. . The required cur e i tersectio is show in the ﬁgure by its elliptical projections on which the pomts s and t

re ne b n o f aux i ar a marked . The curve is Obtai d y drawi g a series il y pl anes throu gh th e vertex o f th e con e an d perpen di cul ar to th e fron t

an e n pl (the horizo tal traces wil l be perpendicular to the X axis) . i n n e e en s and Each auxiliary pla e cuts from the co e l m t , from the

n n s ra h ine I n o ne of give pla e a t ig t l . the ﬁgure , these auxiliary

n b A and F A n pla es is marked y its traces H . This auxiliary pla e cuts from the cone the elements a v and b v and from the plane

n n n f n o f a straight li e m n . The i tersectio s o the projectio s this last lin e with the projections of a v and b v give respectively the points

3 and t on n f n i the required li e o i tersect on.

The curve of intersection of an oblique cylinder with a given

n n b n n n s pla e is fou d y the same ge eral method , usi g auxiliary pla e that will cut elements from the surface o f the cylinder and straight

n lines from the given pla e .

8 2 A ane whi ch i s tan ent to a iven c ne or a c li n der . pl g g o y ,

m n f i urf n contains an el e e t o ts s ace . o f n If , the , the surface a co e ,

or n on n n n n o f n a cyli der , is rolled a ta ge t pla e u til each its eleme ts

n n t n and has come i to this pla e , the par Of the pla e passed over included between the extreme elements is a plan e surface equal to th i n surface as de e e g ve . The surface thus p sed over is called a v lop men t I n o n n of ff n . rder to determi e the positio s the di ere t ele

n o f n n o f n me ts the surface as they come i to the pla e developme t , it is necessary to locate som e curve on th e surf ace whi ch wi ll de nt s h l ne Or a s e cur e n velopi o a traig t i impl v , upo which the actual

n n c n o ff distances betw ee the eleme ts a be laid . . i 42 n n n a . I n F g. Oblique cyli der is Show The sectio n cut f n z n n n be rom this cyli der by a hori o tal pla e happe s to a circle .

n a n n i n Through the poi t a pla e is passed , cutti g the surface the

n n a c and e. n straight li e through the poi ts , , The curve of i ter ° section when revolved through 90 into the plane of the drawing

1 2 4 Th e c rc eren ce o f h s n 3 5 . is show as the circle , , , , i umf t i c rc i h n h f h e en o f th e c lind r i le s t e actual le gt o t developm t y e . If we wish to develop the part Of the cylinder above the inter

n b n a a n n sectio , we draw the ase li e , maki g its le gth equal to the

r A n n n n n . b o circumfe e ce of the circular i tersectio poi t , , the curve ' n n a of the developme t is immediately located over the poi t , at a

n off o n an n a b in dista ce laid eleme t , equal to that from to the

to . n n p base Other poi ts are take in the same way . To locate

' n n o n devel o ment the n 1 2 a other poi t the curve of the p , dista ce ,

O ff on b n 1 2 and n is laid the ase li e , equal to the arc , the le gth c d s n f is the width at thi poi t o the surface we are developing .

B n n n o f n b n and y co ti ui g the process a series poi ts is O tai ed , the surface included between the curve drawn through these poin ts and the base line is the development o f the surface o f the cylinder

n n n a above the i tersectio with the pla e passed through .

n o f n b n r The method for the developme t a co e ei g simila , it

no x l n ion needs e p a t . The problems relating to developments Of surfaces are con k h m stantl a l ed b w r ers in s eet etal . y pp i y o Of course , theoretic

it no ff n n o f ally , makes di ere ce which eleme t the surface is cut “ n but n out for the developme t ; , practically , it is eco omical to ” n so as to make the shortest seam , u less , however , there are good

n n reaso s for doi g otherwise .

8 4 To ‘ he n erse o co th an . t o n ne 3 . PROBLEM 3 ﬁnd t i t c i n f a y wi y

lane p .

ethod n of i n M . Pass through the co e a series aux liary pla es t n n to and ake either perpe dicular its axis , or through the vertex

n n f F r perpendicular to o e of the pla es o projection . o the curve

. o f intersection join the points where the lines cut from the cone

’ n lin c n i tersect the es . ut from the pla e .

EXERCISE S

u of n n o f n Draw the c rve i tersectio a right co e , axis ver

n n w develo a b . ment tical , with O lique pla e Dra the p o f the portion Of the con e between the line o f inter

section and the vertex . 42 n . n of n Taki g the data of Ex , determi e the curve i ter section o f the cone of rays of maximum intensity with

the plane Of the desk .

Make the drawings for the patterns o f the bath- tub

in o n 1 4 shown the ﬁgure page 3 . — Suggesti on This exercise requires merely the devel O pment of portions of tw o cones with vertices at n and

o and n o f n . As i b , the patter s pla e surfaces sume su ta le

n n not n in and in dime sio s for those give this exercise ,

those that follow . Draw a pattern for the sheet metal for a regularly ﬂaring (coni cal) roof connection lik e the o ne Shown in the

e The a s o f c li ndri cal 1 e is ﬁgur . xi the y p p vertical and

the n o f n b R 0 pla e the roof is represe ted y ( , A conical tower is to be placed on the right- hand side

f r l hi n in 82 o the ir egu ar p roof show Ex . . Show the

to be in f true size of the hole cut the roo . — Suggesti on Find the true size of the curve of inter section by revolvi ng the curve about o ne o f the traces

f n n n n . Ar o . t . 53 its pla e i to a pla e of projectio Cf . Draw the pattern for a boot to join two pieces o f stove

one o f is and pipe , which circular the other oval .

l 6 n in and The circu ar pipe is i ches diameter , the oval

n b 2 1 n pipe is represe ted y parallel sides , } i ches apart

and 5 n n wi m n f 4 i ches lo g , th se icircles at the e ds o

I n - dim he . en t parallel Sides other words , the over all

ns f n 2 n Mak sio o the oval are 8 i ches by k i ches. e the “ ” 10 n n . boot . i ches lo g

8 . R To nd the i ntersecti on o an c li nder with 4 P OBLEM 35 . ﬁ f y y an lane y p .

ethod n of i n M . Pass through the cyli der a series auxil ary pla es ,

n n o r and take either perpe dicul ar to its axis , parallel to its axis

n n f n f n Fo r r o f perpe dicular to o e o the pla es o projectio . the cu ve intersection join the points where the lin es cut from the cylinder in in n n tersect the l es cut from the give pla e .

EXERCISE S

Find the true size and shape o f the hole cut fo r a circular

chi mn ey in a tin covering fo r the right- hand Side o f the

hi n in 79 . p roof give Ex .

Draw the pattern for the inclined end o f a bath- tub with a

— n n n n semicircular cross sectio . (The i cli ed e d is as

be n sumed to a pla e surface . Find the true size of the Opening to be cut in the wall

n in Fi 51 3 in . n Show g for a pipe feet diameter , maki g

° ° ‘ n o f 30 and 20 res ectivel with z n a gles p y the hori o tal ,

and n b o f with the vertical pla e Of the ack the wall . I n the ﬁgure a ﬂue from a boiler- house is Shown pass

in o f l F n g through the roof a sma l Shed . i d the true

w in siz e o f the hole that as cut the roof for the ﬂue . Make a pattern to Show a Sheet o f tin to cover this roof

D h in with the hole cut o ut for the ﬂue . evelop t e

clined portion o f the ﬂue above the roof . (Select suit able data fo r the co Ordinates o f the points that are in marked the ﬁgure . )

8 . R E To nd the o ntersecti on o an sur ace 5 P OBL M 36. ﬁ f i f y f o revolut n w h an lane f i o i t y p .

ethod f n o f M . Pass through the surface o revolutio a series auxil iar n n to i n y pla es perpe dicular its ax s . These pla es cut circles from

n n curve of i tersectio .

EXE RCISE S

n 138. F n n o f n in i d the i tersectio the torus give Ex . 1 16 with

an b n O lique pla e . 139 i . Draw the project ons of the hexagonal nu t Shown in the

n n n o f n ﬁgure , represe ti g accurately the li es i tersection .

“ 4 o f b en 1 0. Draw accurately the curves the stu d o f a connect

i n - g rod .

14 1 b n n fan n . The lades Of a ve tilati g are pla e surfaces attached

hub n to . n a Spherical Show the i tersectio s .

8 intersecti n of an tw curved surfaces is n 6. The o y o fou d by pas sing auxiliary planes to cut from each surface lines that can

n n n n be easily co structed . The i tersectio s Of these li es give

f in n l points o n the required curve o tersectio . The auxi iary planes should be selected so that the lines cut f rom the surfaces

n to n o f n o r are straight li es , circles parallel a pla e projectio , some

f ns can be n other curve o which the projectio easily draw .

8 wi n tw o n n n n 7 . A shaded dra g of i tersecti g co es is show

43 n n i n n i . In F g. The same co es are show orthographic projectio

i 44 o f n c n n oh in F g. . The curves i terse tio that are show were tained by the use o f auxiliary planes passed SO as to cut elements

n in n from each co e ; other words , the auxiliary pla es were passed

' n n through the vertices Of both co es . The eleme ts cut from each cone by an auxiliary plane are drawn through its intersections with the base and through the vertex .

Fi 44 an r n A is . n I n g. auxilia y pla e , , marked Si ce it is a

o f b n n plane passing through the vertices oth co es , its horizo tal trace H A is drawn through the in tersection o f the lin e v w with , , " b n b x . the plane of the ases , marked O serve that the horizo tal projections O f the con es are represen ted o n the plane of the bases n d di xi i n and that now the li e through , , is used as the ax s Showi g the intersection Of the horiz on tal and front planes . The auxiliary

n v n c v and d v plane cuts from the co e with vertex , the eleme ts ;

w n a w and w n b . and from the co e with vertex , the eleme ts

The intersections of the horizontal projections of these elemen ts ” n n n n two and two , determi e the horizo tal projectio s Of four poi ts

n in n c n on the curves o f intersectio . The curves the fro t proje tio are Obtain ed by drawing the projections Of the elements Of one

n and n on n in n co e , projecti g them the poi ts the horizo tal pro n n f jections of the curves of in tersectio . The projectio s o two m an n d . poin ts in the curves are marked

e o i ntersecti on o two co es M 37 To nd the curv n . 88 . PROBLE ﬁ f f n ne n Method . Pass through the co es auxiliary pla s draw through

the n omm r n n b li e j g thei vertices . Eleme ts cut from each co e y

these planes determine by their intersections the required curve .

EXERCISES

Find the curve of intersection of two right cones with axes

n ot n e one n n that do i t rsect . The axis of co e is horizo

other vertical n tal , the . Draw the developme t of the

n ‘ n n o f co e with the horizo tal axis , showi g the curve

n n i tersectio . Draw the curve o f intersection o f two oblique cones with

n n n of one o f the i tersecti g axes . Show the developme t

n co es .

8 n n n I Fi . 45 m 9 . g the method most com onl y used for ﬁ di g

n n ~ n n the i tersectio of two cyli ders is shown . The solutio is a si mpliﬁed method o f passing planes perpendicular to the front — plane o ne set parallel to the axis o f the smaller cylinder and

n n a other set parallel to the axis of the larger cyli der . The ﬁgure

n n n is lettered so that no other expla atio is eeded .

0 B LE M To d the cu rve o i ntersecti on o o . PR n tw 9 O 38 . ﬁ f f

s cyli nder .

ho n tw o Met d . Pass through the cyli ders sets of auxiliary

n o ne set . n pla es , parallel to each axis The eleme ts cut from each cyli nder by these planes determine by their intersections the re quired curve .

EXERCISES

D raw accurately the lines o f intersection appearing on the

e o f n n n in surfac the fla ged pipe fitti g , show the figure

144 n u o n page . The ﬁtti g is made p of two cylinders

n n m with their axes i tersecti g each other sym etrically . Find the in tersection of tw o oblique cylinders with their

z n n bases in the hori o tal pla e . A vertical steam- drum is to be put o n a horizon tal cyl in

rical b M n o f d oiler . ake a patter to show the size the

in Fi hole to be cut the boiler plate . (Cf . g.

4 F n n n of tw o n n b n 1 7 . i d the i tersectio cyli ders formi g a ra ch “ b n in n 14 Y fo r a lowpipe as show the ﬁgure o page 6 .

D n o n n evelop both cyli ders t show the curve of i tersectio .

i 4 o f n n n 1 I n F . 9 . g 6 a very simple case the i tersectio of a co e

lin r i n n n n and a cy de s show . The horizo tal projectio of the cyli der s shows immediately the horiz on tal projection of the curve of inter

f h n b n n and the f nt n o t e . sectio . ro projectio curve is fou d y projectio

Horizontal lines are drawn through the fron t projections of the

in a b and c to n n ue po ts , , show a simple method for ﬁ di g the tr

n hs of n n and n he le gt eleme ts of the co e , the true dista ces from t

f n on an n n r vertex o poi ts eleme t . These true le gths a e needed for a. development of the surface o f the cone .

n This is a special case . The ge eral method is stated in the

roblem next p .

To nd the cu rve o i nterse P O B LE M 39 . ct o o a nder 9 2 . R ﬁ f i n f cyli

and a cone.

ethod r o f n n M . Pass th ough the vertex the co e auxiliary pla es

f n The parallel to the axis o the cyli der . elements cu t from each surface by these planes determine by their intersections the required

curve .

n the a b it adv Whe two surf ces have circular ases , is most an

a eous to use a i n u c t g ux liary pla es which cut circles from each s rfa e.

E XERCISES

A circular tower has a conical roof through which a hori

l F n z onta pipe passes . i d the size of a covering for the roof and show in the development the hole cut for the

pipe . Find the intersection o f an oblique cone with a right

do n n n . o t s and n cyli der The axes i ter ect , o e axis is

and n l n n parallel the other perpe dicu ar to the fro t pla e . Make the necessary patterns fo r an arch stone of a conical

a arch in a circular w ll .

A horizontal steam pipe 117 ; i nches in diameter is in tersected by a conical noz z le and two smaller vertical pipes as

n in s10 e o f show the ﬁgure . The p the curved surface of

n z 3 the oz le is 7 5 . (a) D etermine the developed true size o f the hole cut in

z n t b n z the hori o al pipe y the o zle .

b n n ( ) Develop the co ical surface of the ozzle .

52 n nt n n and n n 1 . Fi d the i ersectio of the co e the cyli der formi g

- the steam exhaust head shown in the ﬁgure o n page 150.

40 To nd the li ne o i ntersecti on o an two B LE M . 9 3 . PRO ﬁ f f y s a es o revolu ti on s her e elli soid toru s u rf c f ( p , p , ,

u n n t. Method . If the axes of the two s rfaces of revolutio i tersec the point of in tersection of the axes is taken as the center for a

o f n series auxiliary spheres . These pla es cut circles from each

n n f surface . The i tersectio s o these circles with each other are

n on poi ts the required curve .

n ot n If the axes do i tersect , pass through the surfaces auxiliary planes perpendicular to the axis of o ne surface and cutting circles

f and n rom that surface , cutti g some other curve from the

n n n n n f surface . The i tersectio s of the correspo di g projectio s o the

n curves give the points to be fou d .

n f Th i 47 n o an . e F g. shows the i tersectio a sphere with ellipsoid point m is taken as the center for the auxiliary spheres (at the in

i n tersect on of the axes) . The arcs of the auxiliary circles show de

n s n n xf B n n i n s termine by their i ter ectio the poi t . y co ti u g thi construction a series o f poin ts is obtained which when connected

n o f n n at bf by a smooth curve gives the li e i tersectio . The hori

n b n z ontal projection is obtai ed easily y projectio .

EXERCISES

o f n n o f a s h e and an b e 153 . Draw the curve i tersectio p er o liqu

n co e .

4 in n n of and the s 15 . Draw the l e of i tersectio the ellipsoid toru

n 1 2 shown in the ﬁrst ﬁgure o page 5 .

Suggesti o n Use th e approximate meth od o f tol rcu l ar arcs f or constru cting th e elli se i n th e u re ma i n D A B ' E h Oh F and G 0h p ﬁg , k g C F O h h o H rE o .

n n in b l Th nt r i e ﬁtti . e 155 . A is show the ﬁgure e ow ce e p p" g line of o ne portion is the arc o f a circle intersecting

n i n the axis of the smaller portio wh ch is cyli drical .

1 7 n and 1 in Make the diameters respectively } i ches ch . Draw the curves o f intersection o n the outside in of the ﬁtt g .

The sewer shown i n the ﬁgure below by its section is inter sected by two elliptical sewers with major and minor

4 and 2 n n axes o f feet % feet . The dime sio marked r

i s 5 in lan n for the large sewer feet . Show a p drawi g the lin es o f intersection to be made in the mason ry where b f the sewers meet . The ottoms o the three sewers

n s on are in the same pla e . The small ewers are o p

osite of . p sides , the large sewer

CHA PTER V

MISCE LLANE O U S PRACTICAL E XE RCISES

' A shaft o f a mine follows approximately the line between

n a — 3 an d b - 1 the poi ts , , — A tunn el is to be made from a poin t e 1 §

on n n n F n the side of a mou tai to i tersect the shaft . i d

n o f nn and n be the shortest le gth the tu el , the a gle tween the center lin e o f the shaft and the cen ter line

nn of the tu el .

n Fi 48 nn and an I —b n as I g. a cha el eam are show they

n n o n intersect i a roof . Take the followi g co rdi ates fo r the poin ts marked in the ﬁgure : a

b _ 4 C 5 _ 4 d ) ( + % 7 — — a e w 2 l w 4 n e , s ) , ( e, , i ) , i ( e, 4, u) ,

- 1 — 1 - 3 2 7 O h 2 n (9 4 35 , 5 , ) , , Draw the projectio s o f a bent plate conn ecti on and in a section drawing

n b n Show the true a gle etwee its sides .

n 2 in A cyli drical pipe , feet diameter , passes through the

n in Fi 14 at n in n n roofs show g. a poi t the i tersectio

o f n o f the roofs . The axis the pipe is perpe dicular ° to the line o f in tersection and is in clined 45 from the

F n o ut o f vertical . i d the true size of the hole cut

the roofs for the pipe . Obtain by developmen t the true siz e and shape o f the coverin g needed for a symmetrical dome with eight

sides . n b n Fi 49 n . I g. a portio of a locomotive oiler is show Make the to p elemen t o f the slope sheet at an in clina ° n o f 30 z nt and srz e tio to the hori o al , show the true

be in n o f a steel plate to used maki g the slope sheet . The sand —box on a locomotive stands partly o n the slope

on in n F n sheet and partly the cyl drical portio . i d

the shape of the bottom o f the sand - box to ﬁt the 1 1 n 6 . boiler draw for Ex .

n ni n o f 163 . Make the patter for the co cal portio the eave

n in tro ugh outlet show the ﬁgure .

b f - 164 . A lock o wood with a square cross section has been

n in n in tur ed a lathe to the shape show the ﬁgure . Show the lines of in tersection between the part with

n and n the square sectio the co ical portion .

— ’ Make the patterns for the oil can and grocer s scoop

n o n 1 show page 60. — The stack of a boiler is supported by gusset plates as

n in Fi n 50. show g. Make the patter s for the gusset

plates . Find the intersection o i a sphere with a cylinder whose

n n axis does o t pass through the ce ter of the sphere . Fin d the in tersection of the cylindrical ceiling o f a corridor

with the hemispherical ceil ing of a vault. DO ME4

DEVEL PMENT O F T HE SLO PE SHEE O T . F 49 I G . . e l 1 7 A s micircu ar wire loop, } n in i ches diameter , rotates on a vertical axis supported at the poin ts a — 1 , and b —1 — 1 ’ , f}) at the ends o f its vertical

. b n i n diameter A all , i ch diameter , is attached to the end f o a horiz ontal supporting arm and revolves about the point c The distance from the n of ce ter the ball to the center of revolution is 1 i ch n . Will the ball meet the wire loop ?

an A metal shade for electric lamp is made up of a hemi

and o f n sphere half a circular cyli der . The axis o f the n n o f the cyli der passes through the ce ter Sphere . D raw n r n the i te sectio of the two surfaces and a pattern to be used in cutting o ut the metal to make the shade . F n n n i d the i tersectio of a hexagon al prism with an oblique

n . D pla e raw the development of the part o f the prism b n o f n a ove the li e i tersection .

2 n ne 17 . Make a patter for o of the sections o f the stovepipe

b n in el ow show the ﬁgure .

I n Fi ; 51 b in n T g a uttress a slopi g wall is shown . he

i n n c . n 10 and 8 d me sio s a d 9 are feet feet respectively .

As b n n an sume other suita le dime sio s , d make all the

patterns a stone- cutter wil l need for making the top

n b sto e of the uttress .

’ Make the patterns for the furn ace- setter s offset boot

n in n show the ﬁgure . The sectio of the top portion is

and r n oval of the lowe portio circular .

175 F n t m be cu t in . i d the size of the hole tha ust the roof of

h in Ex 137 to t t b t to r n n t e shed . allow a igh el u betwee

he and r t pulleys q .

n ash - n in 1 2 1 . F 5 . 76 Make the patter s for the chute head show g.

D n in n 1 77 . evelop a portio of the vertical pipe the accompa y ing ﬁgure to show the true siz e of the hole to be cut so ° n n 30 that the smaller pipe , i cli ed to the vertical , may B e n . n b joi ed to it oth pipes have circular sectio s .

n n n nn n b n 178 . Make the patter s for the tra sitio co ectio etwee

n u a square duct a d a circ lar pipe .

CHAPTER VI

SHAD E S A ND SHAD OWS

A er n who n n 94 . p so u dersta ds descriptive geometry can usually get a correct conception o f an object from the orthographic pro

i ns o f n n ject o its edges or other li es of its co tour . A proper c onception of the form and relations of the different parts of some

o f n o bjects requires , however , more careful study the projectio s

f r r For n than is desirable o practical pu p oses . this reaso drawings are sometimes made to show an effect simil ar to that produced by

n n ff is the shadows from illumi atio . This e ect a great assistan ce also in making drawings plain to persons who are un familiar with

n the methods of orthographic projectio . The subject o f sh ades an d sh adow s treats o f the application o f the methods qf descriptive geometry to produce the effect o f l in n n n i lum atio , which gives a more real appeara ce to the projectio s o f n b a o ject .

an b n t th e If Opaque ody is placed ear a source of light , par of

i be r and n be surface w ll b ight the remai der will dark . A portion o f the light from the lumin ous body wil l be inter cepted and a

n b n b y be in n i portio of the space ehi d the od will dark ess . Th s

e sh ad w o f b and n o n dark space is call d the o the ody , the li e its surface separating the bright side from the dark is the sh ade

lin e.

. I n Fi 53 a n n n n 9 5 g. co e is show ear a vertical pla e P . Paral

n s n n a b c d lel rays of light are represe ted pas i g through the poi ts , , , , v n r n o f i l n mi n and . The i te sectio s these rays w th the p a e deter e

sh adow o f n o n n sh ade n e is n the the co e the pla e . The li fou d by

n n ‘ n the u n n joini g poi ts o s rface where ravs are ta ge t .

6 i he o f . n is t t t 9 U less it o herw se speciﬁed , rays ligh are to

n B n n be represented by parallel li es . y the usual co ve tio n in practical drafting the rays are taken as coming over the left

so n and n n o f shoulder , that the horizo tal fro t projectio s a ray ° ° n o f 4 5 and 1 h make respectively a gles 3 5 wit the X axis .

n n 9 7 . The method for co structi g the shadow of a cone o n the

i n n b n in Fi 4 hor zo tal pla e through its ase is show g. 5 . The prof

ectio ns v n and the j of a ray through the vertex , , are ﬁrst draw , in tersection o f this ray with the plane of the base is found at the 1 2 n b the ns v and v L n s n in poi t marked y projectio . i e are draw 1 z n n n n b and v the hori o tal projectio ta ge t to the ase through . These are the limiting lines of the shadow and determi ne at the poin ts

n n n e v and d v o f ta ge cy the shade li es . “ 5 Fig. 5 shows the shadow of a circular cap o n a cylindrical

an n n n n . R colum . Scarcely y expla atio is eeded ays are drawn ” through points o n the cap and their in tersections with the sur

n n n i face o i the colum are easily fou d . The horizo tal project ons

i n n n n xer ical of the rays determ e the poi ts of i tersectio . The t

n ti and 7 n b n n shade li es at 3 are determi ed y ta ge t rays .

8 . an o n z n n n 9 The shadow of ellipsoid a hori o tal pla e , P , is show

i li ne o shade o f n n F . 5 . w i g 6 The f is made use to determi e the shado . “ ” Obviously it is the line of contact with the ellipsoid of a cylinder

n n n n . of rays ta ge t to the surface This li e of co tact is , of course ,

I n n an i n . n a ellipse the ﬁgure two projectio s of ellipso d are show .

’ n A n the c n If a vertical auxiliary pla e , , is draw parallel to dire tio

‘ o f and n 0 o f the ; c ut an the rays through the ce ter , , ellipsoid , i will ellip se from the ellipsoid and two elements tange nt to the ellipse “ ” n n n w b from the cylinder of rays . These poi ts of ta ge cy ill e the

n in . If i highest and lowest poi ts the ellipse of shade the pla ne . 4 s revolved about a vertical axis through 0 so that it is parallel to

n n n b the fro t pla e , the sectio cut from the ellipsoid y the auxiliary

n plan e will be made to coincide with the front projectio . A ray drawn through the cen ter 0 intersects the horizontal plane P at the ' 2 n n an 0 A n t poi t marked by the projectio s o d . fter revolutio wi h h the auxiliary plan e this ray is shown by the projections o 0 1 an d 7 0 02 D raw the fron t projections o f rays parallel to tangent

n f f an f n to the revolved sectio o the ellipsoid at al d bl . Whe the

’ plane A is revolved back to its former position these tangent

d b I n n n i f n a an . n a b poi ts are at the fro t projectio , the , is the

I n n major axis of the ellipse of shade . the horizo tal projection o f h h a b n and the ellipsoid , is the mi or axis of the ellipse of shade , h h h h a b e i perpendicular to is the major axis .

o f o n n n The shadow the ellipsoid the horizo tal pla e , P , is the shadow cast by a section of the ellipsoid in cluded by the ellipse of 0 ﬁnd n n n shade . T this shadow determi e the i tersectio s with the

n o f n 0 a b e and n pla e , P , rays through the poi ts , , , f . These poi ts ’ ’ ’ ’ n and n a b an d e determi e the major mi or axes , f , of the elliptical shadow . EXERCISES

F n an o n n i d the shadow of ellipsoid a vertical pla e .

F n a o n n b and i d the shadow of a sphere , ( ) a pla e elow

the z n n b o n an b n parallel to hori o tal pla e ; ( ) o lique pla e .

D n o n raw the shadow of a dormer wi dow a roof .

o f i n o n D raw the shadow a ch m ey a roof . The stair ramp shown in the ﬁgure is parallel to the slope

f n n n o f is 1 o the steps . The i cli atio the stairway ver

2 z n ns o tical to hori o tal . Co truct the shadow f the

I' o n s m p the step .

nd n in Fi 55 Draw the cap a colum g. with the axis

n hori z on tal instead of vertical . Show the shade a d

shadow . Make a simple drawing o f a section of an engin e cylinder

n n R n i n through the ce ter li e . eprese t the whole p sto in the cylinder by showing the shadow o n the inside o f the cylinder o f the po rtion that projects beyond the

e n s ctio .

CHAPTE R VI I

WARPE D SU RFACE S

W ar ed surfac s n n 99 . p e are disti guished from pla e surfaces and n n n and n b surfaces of si gle curvature (co es , cyli ders , co volutes) y

f n n e mn I n a di fere t positio of the le e ts with respect to each other .

b n n be all the surfaces that have ee discussed , a pla e could always

' at tw o o f n n passed through least the co secutive eleme ts . This

b n b I n property was very servicea le for the solutio of pro lems .

e no n n in warped surfac s , however, two co secutive eleme ts lie the

n t is lin s d aw n h h an tw o n ec e same pla e ; hat , e r t roug y co s utiv position s of th e straigh t lin e generatin g th em are n either par

allel n o r in tersectin g . Examples o f warped surfaces are shown in

a 5 b n F 57 7 a 58 . igs . , , d

A War ed sur ace be n b a s trai ht p f , therefore , may ge erated y g

n n so that a w a s t u ch es tw o en lin es an d re ai n s li e movi g . it l y o giv m

n l * paral lel to a gi ve p an e . The ﬁxed lin es which the moving line touches are called the di rectri ces and the plan e is called the pl an e

i r d ector .

1 h erb c arab l id 0 0 . The yp oli p o o is a warped surface with a plan e director and two straight lin e directrices which are no t in

n n the same pla e . It takes its ame from the fact that curved sections made by planes cutting the surface are either hyperbolas

Fi A s « o r parabolas (cf . g. the directrices approach paral

l m n as i i n le is the surface approaches a pla e a l m ti g surface . This

f n as in n surface is o some practical importa ce , it is used maso ry

n nd — o f Fi n a . 60 co structio , the cow catcher a locomotive ( g ) is f usually o this form .

h ere is a reat ari et o f w ar ed su rfaces al l diff eri n i n th ei r mo T g v y p , g de o f gen erati on an d pro perti es ; but this expl anatio n is suffi cien t fo r the surfaces

th at will b e discussed .

I n b b an n n a hyper olic para oloid , y pla e parallel to a pla e director

out in n and n n n will each directrix a poi t , the li e joi i g these two

n be an n o f n poi ts will eleme t the surface . The eleme ts may be regarded as lying in a series o f planes which are paral lel to the

n an d n n pla e director dividi g the two directrices proportio ally .

n an n no t in n i If, the , y two straight li es the same pla e are div ded

n n n in n n i to proportio al parts , the straight li es jo i g the correspo d ing poin ts o f division are elements o f the surface o f the hyperbolic b 9 para oloid .

1 1 o f i c n 0 . The characteristic properties th s surface a be most convenien tly investigated by referen ce to a pictorial drawing

nd n I n Fi 61 showing the directrices a the pla e director . g. we

n n n n shall take for simplicity of represe tatio , the horizo tal pla e

lane di rector n a c nd b d marked H for the p . The li es a shall be

i rectri ces F n o n n n o f the d . rom poi ts these li es the eleme ts the

r n to n H a b and c d ae . surface , , draw parallel the pla e director,

n ns n i a c Fo r the ecessary co tructio , draw through the directr x

n V b d and n n a vertical pla e , 1 , parallel to cutti g the pla e director

117 n n n V in the line x1 1 . The draw a other vertical pla e , 2 , parallel

b d n n H in n x n to VI through , cutti g the pla e the li e 2 The li e c n n xl x1 and d g n n f is draw perpe dicular to , is draw perpe dicular nd c d be a . to x2 x2 ; therefore i g will parallel equal to Through

n n i o n a c i n H and a y poi t , , draw j parallel to the pla e director , , n o f cutting b d in j ; then i f is an other eleme t the surface . The

n n x x and i t n to line i h is draw perpe dicular to 2 2 , perpe dicular

k be a an d i . A x1 ; then l will p rallel equal to i lso ,

1 235—4 } ' c l f

a t b h Therefore and I t follows that l ie and f 9 cut a b In the

n V n a b in an same point r . Now draw a pla e parallel to I , cutti g y

n n n n n i H i be to xl x1 and poi t , the its i tersectio w th w ll parallel , l i n o — h In 0 In m and - 0 i s and W l ll cut l , j g , also 9 parallel 5771 c

and to a equal to ti and m p is parallel equal i . Therefore

n is n n n which proves q p a straight li e , i tersecti g the

n a b i and c d and n n ro three eleme ts , j , , dividi g these eleme ts p

n so t portio ally , tha

a n i c q= p ' n b q j p d

If the elemen ts a b and c d are taken as directrices and the vertical

n as n b n be pla e V1 a pla e director , the surface which is o tai ed will iden tical with that having a c and b d as directrices and the plane

a b i H n i . n . as its pla e d rector The eleme ts , j , etc , are called

n o f ﬁ s en e a i n and o f set a c eleme ts the r t g r t o , those the other , as ,

n V n . e p, etc , which are parallel to the vertical pla e , I , are call d

n sec n d en era n . n n eleme ts of the o g tio We have show , the , that

n e n n n every eleme t of eith r ge eratio i tersects all those of the other .

i dou bl ru l ed has two sets of Th s surface is y ; that is , it

- n n s and an n o n straight li e eleme t , through y poi t the surface two

- n n n straight line eleme ts can be always draw . Each eleme t o f o ne n n n n o nt o w n set bu t ge eratio i tersects eleme of its , meets all

n o n r in i n F . 5 the eleme ts of the other set . The li es the su face g 8

h d n nl show t is ouble ruli g very plai y .

1 2 n f o f 0 . Eleme ts o the surface a hyperbolic paraboloid are

n in F 62 and n . 63 . a b and c d represe ted igs Two straight li es , of de ni te n and n ot in n n t ﬁ le gth , the same pla e , are divided i to he

n n n n n same umber of equal parts . The li es joi i g correspo ding points

f n n f An n o divisio are eleme ts o the surface . eleme t marked my

n in b f n is show each ﬁgure . O serve that the di fere ce between the two ﬁgures is merely that the same points of division on a b

n d c d n f I n Fi 2 a are joi ed up di ferently . g. 6 the method of joining the poin ts of division is the same as shown in the pictorial drawi ng

in Fi 1 6 . g.

' rec rices o f war ed s r ace in 1 9 3 . If the di t a p u f gen eral are no t

in l n n n limited e gth , it is ecessary to have give the directrices o f “ both systems of ruling o r the directrices and plan e director o f

in i l F . 4 n t 6 . n r o e system . This las case is i lustrated g The li es q and s t t and th e n n b are the direc rices , pla e director is give y its

H F an n n and . traces P P To draw eleme t of the surface , the ,

s t some point as e on the directrix is assumed . Through this point

n d b and c d a b and c d draw two li es l l l l parallel to , which are

n any two lines in the plane director . The i tersection o f the plane o f the tw o lin es through e with the directrix q r is a point in the

n n n can e n required eleme t of the surface . This i tersectio b fou d very simply without ﬁnding the traces o f the plane of the lin es

n d F n r n er en a l bl a d cl l . irst pass through the li e q a pla e p p di cular to the front plane and ﬁnd the intersections o f the lines

n n n f a a ; b1 and cl dl with this pla e . The i tersectio o l bl with

i n n n in n th s pla e has , of course , its fro t projectio the fro t trace at

f in n and n n n . g , a projecti g li e determ es the horizo tal projectio

' i c d intersects the n r and n o f S milarly , l l pla e of q at f, the pla e d n r a he n a b and c t i . t li es l l l l i tersects the directrix , q , The line c i is in a plane parallel to the plan e director and touches the

an n two directrices . It is therefore eleme t of the surface through

n e B an n of n can an assumed poi t . y this process y umber eleme ts be drawn .

1 0 one n o f n o n an 4 . If projectio a poi t y warped surface is

n n n n be b n k ow , a other projectio may located y drawi g through the given projection a lin e perpendicular to the plane of projection

n n n in which the give projectio is represe ted . Through this per pendicul ar line pass a plane which will in tersect the elements o f

i n n n n n o n the surface poi ts which , whe joi ed , will give a li e the

n n n n surface . The i tersectio of this li e with the other projectio

n n n o f n of the perpe dicular li e is the required projectio the poi t .

h e b l d o f re n 1 0 5 . The yp r o oi volutio is a warped surface gene rated by a straight lin e revolving about an axis no t lying in the

n i n n same pla e w th it . Usually a vertical li e is take for the axis . Projections o f this surface showing a number o f elements are

in Fi n n 65 . represen ted g. The method for co structi g this surface

Fi a b i An n n 66. s . is shown i g. The axis vertical eleme t of the

m n en era r x b can be surface , as , is called a g t i , ecause the surface

n n in generated by revolving this li e about the axis . Each poi t m ib z n The n as it revolves about the axis descr es a hori o tal circle .

f n in n t n true radius o each circle is see the horizo tal projec io . Thus the poin t r o n the gen eratrix n earest the axis describes the circle of which the diameter is shown i n the fron t projection of

f rf vf b b n the surface as u . This smallest circle descri ed y a poi t

i s in the generatri x called the ci rcle of the gorge. The diameter

b b n f of the circle descri ed y a poi t p is gl (M, which is equal to h h 2 > < b p

b o f n doubl ru led as The hyper oloid revolutio is a y surface ,

sets f - n n ne n f it has two o straight li e eleme ts . O eleme t o each set

n n n n o n A n n f ca be draw through a y poi t its surface . eleme t o one gen eration can be produced to in tersect all those of the other

n n i n n ge eratio , for the s mple reaso that they lie upo the same sur

n an n n n n face . If , the , y three eleme ts of either ge eratio are take

as an e n o f n n b e n directrices , y leme t the other ge eratio may take as

n n n a ge eratrix which , whe movi g , will produce the same surface . The method for drawing a plane tangent to a hyperboloid o f

n in Fi 6 n n in n 7 . revolutio is show g. The pla e , P , show the ﬁgure ,

n n n e i f n o ne n is ta ge t to the surface at the poi t . o ly projectio Of h 0 is n sa c an n the d 6 give , as y , draw eleme t of surface through

and ﬁnd cf b n n n d e O ne c y projecti g to the fro t projectio of . o f the other set of elemen ts passing through the point is shown

n n the n an n t by f g. The two eleme ts determi e pla e t ge t o the

n e surface at the poi t .

6 A h eli coi d r screw s r a e 1 0 . o u f c is a surface generated by a straight lin e moving so that it is always touching two heli cal di rec

n n n n I n n . F an trices lyi g upo co ce tric cyli ders igs . 68 d 69 two cyli n

n n . O n i n o f ders of revolutio are show each is a port o a helix . The straight lin e m n generates the surface o f the helicoid I n both

f n in th ﬁgures . The di fere ce is merely the slope of e generating

n b o f n line . Whe oth the directi g helices of a helicoid have the

same i tch it u ni or m, i tch s p is called a helicoid of f p . If the pitche

are not the same h of var i n t the surface is called a elicoid y g pi ch .

tw o s n in F 6 and The example of helicoids of u iform pitch igs . 8 69 diff er only in the relative position o f the moving o r gen erating

n n n n line . I both cases this li e moves u iformly arou d the vertical

n on n axis , while , at the same time , all the poi ts it move u iformly

n I n n b in a directio parallel to the axis . either ﬁgure the poi t in the generating lin e m n will describe the helix a b c d lying o n the

nn n and in m and n b of i er cyli der , the po ts will descri e helices the

n same pitch lying on the outer cyli der . These helices traced by

n c and h n n m a d n are marked respectively i g i j . The ge erati g

i n m n in n n n nn n l e will always lie a pla e ta ge t to the i er cyli der , and will in tersect the vertical element at the point o f tangen cy

n n n at a co sta t a gle .

I n o f Fi he n n m n . 68 t n the helicoid g ge erati g li e , , is always

n n of nn n and is perpe dicular to the vertical eleme ts the i er cyli der ,

‘ ri ht heli coid Fi 9 n n called a g . The helicoid of g. 6 has the ge erati g

n m n n n n nn n li e , , i cli ed to the vertical eleme ts of the i er cyli der ,

nd an obli u e heli coi d a is called q . f n n o er endicular . to Whe the eleme ts a helicoid are p p the axis ,

- i s s o f Fi the surface the same a that a square threaded screw ( g.

and n th e n is V— r Fi whe a gle is less it that of a th eaded screw ( g.

Fo r the V- thread of the Un ited States standard screw the angle between the elements and the axis is

1 Fo r n n 0 7 . a satisfactory represe tatio of a helicoid it is not

n b n necessary to draw a large um er of eleme ts of the surface . Usu ally it is shown better by drawing a few elemen ts for a small portion

I n Fi 72 an b wn n o f the surface . g. o lique helicoid is sho resembli g

i 9 o rtio o f n re r the o ne in F g. 6 . The small p n the outli e that is p e sented is easily recogn ized as like that in the groove of an auger

- or a twist drill .

o n i sa n If th s surface it is required to locate , y , the fro t pro

ectio n o f n n z n n n j a poi t whe the hori o tal projectio is give , draw

n n n o f the hori z o tal projectio of the eleme t the surface at the point . The requi red front projection of the poin t is then found by

h n n n n projecting to t e fro t projectio of the eleme t . If o ly the

n on of a n n n n n fro t projecti poi t is k ow , the horizo tal projectio is determined by drawin g a lin e through the front projection

n n A n n i parallel to the horizo tal pla e . y pla e passed through th s

n n in n n n li e will cut the eleme ts poi ts determi i g a curve . The hori z on tal projection of this curve will in tersect the correspond ing projection of the line at a poin t which is the required pro

i n ject o . If the surface of the helicoid in the last ﬁgure is cut by a vertical

n throu h and n to n n yla e g the axis perpe dicular the fro t pla e , the section cut out will be like the shaded drawing at the right - hand

H o f n i n n is side of the ﬁgure . alf the curve show the sectio marked

3 x n b n n 1 1 y1 . This ﬁgure is easily co structed y drawi g eleme ts o f n i n n n d the surface . The poi ts which the eleme ts are i tersecte b n n n n s y the cutti g pla e determi e the curve of the sectio . Thu an n n n n x n i eleme t i j i tersects the cutti g pla e at . This poi t s located On the curve in the section by projecting hori z ontally f from x and laying o ff from the axis the length 01 x1 equal to h k as n n to i n n o . The other poi ts eeded determ e the curve are fou d in n b n the same way . Such a sectio cut from the surface y a pla e

meri di an sect on through the axis is called a i .

n n n n 1 08 . A pla e is draw ta ge t to the surface of a helicoid at a given point by drawing an element through the given point and a tangent to the helix lying in the surface an d passing through the

n in tw o l n n t n n give po t . These i es determi e the required a ge t

n As n w no t be n n pla e . the surface is warped the pla e ill ta ge t

n an n and no t n n throughout the le gth of eleme t , will co tai , there

a n n b n n an n fore , ta ge t to the ase at its i tersectio with eleme t

n n passi g through the poi t .

n n n n 1 0 9 . The i tersectio of the helicoid with a y given pla e is

n b n z n n fou d y passi g hori o tal auxiliary pla es through it . These “ ” n e u al to the base and pla es will cut from the surface Spirals q ,

n n n n in n f from the give pla e horizo tal li es . The tersectio s O these

n n o n n n n li es will give poi ts the required li e of i tersectio . The “ ” projections of the different Spirals are no t n eeded if a curve of

n n n the base is made o a tra spare t sheet.

EXERCISES

Locate a point o n the surface o f a hyperbolic paraboloid

n b ne n n n give y two limited li s . Draw a ta ge t pla e

i n through th s poi t . Three oblique li nes are given as belonging to one set of

nt o f b b F n eleme s a hyper olic para oloid . i d three ele

n o f set me ts the other . Show several ) elements of the surface of a hyperbolic

b n b two and n para oloid give y directrices a pla e director . Draw the projections o f a hyperboloid o f revolution and

n n n n n t assume a poi t o the surface . Draw a pla e ta ge

i n to the surface at th s poi t .

A n an b L ssume the projectio s of o lique helicoid . ocate a poin t o n the surface and draw a tangent plane at

i n th s poi t . Take the projections o f the h elicoid constructed for the preceding exercise and draw a meridian section and

n n three tra sverse sectio s .

n a a s - a e b Draw the projectio s of ( ) quare thre d d screw, ( )

a V— a re thre ded sc w .

CHAPTER VIII

P E RSPE CT IVE

I I P E RSP E CT I V E Is n n o . the art of represe ti g objects as they

n no t fi appear to the eye . The pri ciples are dif cult for a student

n n n who u dersta ds the methods of orthographic projectio . The signiﬁcance o f perspective drawing can be explain ed best b n in f n y Showi g what way it di fers from orthographic projectio . I n perspective the eye o f the observer is at a ﬁnite distance from

I n n the object . orthographic projectio the views r epresen t the

b as n n n n n B o ject see whe the eye is i ﬁ itely dista t . y the per

S ective ir n n n p method , the , the li es draw from poi ts o n the object to n and n n o f the eye , co verge i tersect at the poi t sight .

1 1 1 s n n an . A per pective drawi g represe ts object as seen through

n be n n I n . Fi a vertical pla e which is assumed to tra spare t g. 73

b v a b c and ers the eye at S , the o ject (a square pyramid ) , the p pec e n vi? a? bp 019 n n tive proj ctio are plai ly show . The light solid lines in the ﬁgure represent the rays from the corners o f the object to the

n n n n n eye . The straight li es joi i g the i tersectio s o f these rays with

n n o f b the vertical pla e form the outli e the o ject . If this perspec and l tive picture is shaded co ored , it will exactly represent the

‘ n n object as see from the viewpoi t .

The vertical plane o n which the perspective drawing is made

i c ure an and n is called the p t pl e , the positio of the eye is called h the poin t o f sig t .

The simplest construction o f a perspective drawing is obtain ed b ns o f n o f i y the use of two projectio the poi t s ght , together with n n ns o f b T the two correspo di g projectio the o ject . he method is l i n Fi 4 i lustrated g. 7 . The light solid lines show the horizo ntal an d

n n o f n d n z n fro t projectio s a recta gular car , held early hori o tally . Tw o

n a and b n n cor ers are marked . The projectio s of the poi t of sight " and Sf n n n n are S . Dotted li es are draw j oi i g the corners o f the " card in the horizontal projection with S and joining the corners in n n Sf the fro t projectio with . These are respectively the hori z o ntal and front projections of rays from the corners to the point

f F n hori zonta o sight . rom the poi ts where the l projections of

n the n n n these rays i tersect X axis , li es are draw perpe dicular to the axis to intersect sthe corresponding front projections o f the

n n rays . The cor ers of the perspective projectio are thus deter

n and b n n n n mi ed , y joi i g these cor ers the outli e of the card shown

n n Fi by the heavy solid li es is obtai ed . g 75 shows a perspective “ n n b b drawi g of a scree with three lades made y this method .

n a b and c to n n n The poi ts , , are marked make the co structio plai er .

n n n an b in n Usually whe represe ti g o ject perspective , the horizo tal

b n I n plan e o n which the o ject rests is show . the ﬁgure the line marked GL is the fron t trace o f the horizontal plane through the

f n bottom edges o the scree . If a house is to be represented in a

n o f n n b z n drawi g , the level the grou d is represe ted y a hori o tal

n n n n pla e . The fro t trace is the very properly called the grou d l n s na n to n n . a i e Thi me is give , however , the fro t trace of y hori z n al n o n an b n be o t pla e which o ject is imagi ed to placed .

1 1 2 be b h . A picture may made very simply y the met od de

t n be n I n b if no . scri ed , there are ma y details to show practical

n no t n f drawi g , however , this method is ofte used . The dif iculty is that for a picture o f suitable proportions the horizo ntal projectio n h f n S be so the o the poi t of Sight , , must usually located far from X

n o f the n - b Al f r axis that it is beyo d the limits drawi g oard . so o an object with many details the number o f construction lines

n n n n becomes so great as to be very co fusi g . It is the ofte difficult

n be n Fo r s ns n to decide which poi ts are to joi ed . these rea o , the ,

o f n n an abridged method perspective drawi g is commo ly used . h k n b i n a b an af f i 76 n a b t d b . I n F g. a li e is show y its projec o s This lin e is parallel to the horizontal plane and makes an angle ° n n B d t has b e n of 45 with the fro t pla e . y the metho tha e j ust n ' bp ne n a . explai d , its perspective projectio is fou d at The other general method for cons tructing perspective drawings may be

n a b is n n illustrated with this ﬁgure . The li e produced i deﬁ itely , h n in b n n n as show oth projectio s . A projecti g li e from S to the end of the line through a b must be represented parallel to it an d 0 4 h e n n making an angle of 5 with t X axis . This projecti g li e meets the X axis at u ; and th e perspective of a point o n ab at an

n n an n o n inﬁ ite dista ce is at M . The perspective of y poi t the line

a and b b n af and n produced through is , therefore , etwee M I i h n O bserve S t = tu = fM H n ns S . in this co tructio , that orizo tal l es

n n n n but n in maki g the same a gle with the fro t pla e , slopi g the

n n n opposite directio , would co verge toward a similar poi t located

an n . n at equal dista ce to the left of S It follows , the , that all ° h oriz ontal line s at 45 to the picture plan e (front plan e) converge in perspective tow ard poin ts on eith er side o f Sf an d at a distan ce

. from th i s proj ection equ al to th at o f th e poin t o f sight from th e

n in ce the n n pi cture pla e . S locatio of the poi t M depends only . ° o n di recti on a b and n ot o n osi ti on an z n 4 the of , its p , y hori o tal 5

n in n n i rn li e the drawi g will co verge toward M . Th s po t is called

- n the m easurin g poi t .

n n n n in Fi 7 Some dime sio li es are show g. 6 to Show how this

n can be n o ff n . Fo r poi t used for layi g dista ces example , the perspective projection of the poin t b could be found by laying h o ff d n b o n — n o f bf the ista ce from to the X axis the left ha d side , and b? would be found at the in tersection of this lin e with the

n n n n bf nd 7 I n projecti g li e joi i g a S . the same way the perspective h of a point e is found by laying off the distan ce from c to the X — f axis o n the left hand side o f c and locating c? at the in tersection

n cf f n - n ll" of this li e with S . If the measuri g poi t had been located o n - n Sf n in the left ha d side of , the these same cases the distances mention ed would be laid o ff o n the right- hand side of the fron t projections . This method o f cons tructing perspective drawings with the

n - n n in Fi 7 help of measuri g poi ts is show also g. 7 . A perspective

n o f b n b in drawi g a cu e with circles i scri ed its sides is illustrated .

The measuring- point M is located by maki ng SfM equal to the distan ce from the poin t of sight to the picture plane (Sh to the

b n X axis) . O serve that there are a umber of parallel lines in a

n n n cube . The co structio ca be simpliﬁed if we n otice that all “ ” n n n n parallel li es must co verge or va ish at the same poi t . Every

set n n an sh n - n n of parallel li es has the a v i i g poi t . The li es which

n n n are perpe dicular to the picture pla e have , of course , their va ish 0 i n - nt S7 and z n 45 n n Fo r an g poi at ; hori o tal li es va ish at M . y

n n n - n n - n and th e n drawi g the va ishi g poi ts , the measuri g poi ts , fro t

n o f be in n projectio S must a li e parallel to the X axis . The judicious u se of measuring and vanishing—points saves

i n I n Fi b n n . 77 much la or maki g perspective drawi gs . g the

- vanishing points of the horizontal edges o f the cube are at V 1

and V is b n an fo r 2 . Each located y drawi g edge which the dirco

n b n n n n Sf tio has ee determi ed , to i tersect the li e through parallel

to the X axis .

1 i n o f b 1 3 . The perspective draw g the cu e illus trates also a very satisfactory Iii etho d for drawing the perspective projections

n in n of circles . Poi ts the perspective drawi g of the circles are found by locating diagonals o f the circumscribing squares formed

f n n by the edges o the cube . Other li es are also draw parallel to the sides o f the same square through the intersections of the diago h n ns n als with the circle . The i tersectio of these straight li es locate usually enough points to determin e the perspective of

( o nstru ctio n tw o n a and b o f the circle . The for poi ts the circle

in are illustrated the ﬁgure .

— P r c e D s rt n . n f 1 1 4 . e spe tiv i to io The pri ciples o perspective are no t difﬁcult to apply in a mechanical way by those who have no artistic trai ning ; bu t distorted results are obtain ed from absolutely

n n n n b . correct applicatio s whe a surd co ditio s are assumed If , for

is n n b example , a large house represe ted with the poi t of sight a out

n n b twe ty feet from the fro t of the house , o viously a poor result is

n fo r n can be obtai ed . Nevertheless such a case the pri ciples applied

‘ as well as to any other . A perspective drawing should show the object as it appears to

n b n the eye . If is importa t , therefore , that the est viewpoi t is

obtained ; and care in selecting the viewpoint is as essential as a

n o f l k owledge the ru es . If a house about forty feet high is to be

n of be n b sketched , the poi t sight should take a out eighty feet from

n the picture pla e . A good rule to follow is to make this distance

b dimensro n n a out twice the greatest . Whe large objects are to

be n represe ted , the most satisfactory results are obtain ed usually

n n o f is n n in n o f b whe the poi t sight take early fro t the o ject . It is

b n lan n be n n n prefera le , the , that the p drawi g should show i cli ed ) n to the picture pla e .

“ ” 1 1 Pers ec i e S ch s fr — 5 . p t v ket e om Work in g D rawin gs A

' very pro ﬁtableappli catio n of the methods o f perspective drawi ng — “ ” “ ” is found in making free hand sketches from working or shO p

n n in n drawi gs show orthographic projectio . The working drawing represents certain in formation about an object by a

c n n colle ti o of views . Several views are ecessary to represent the

b I n n n n o ject completely . perspective drawi g the same i formatio i wn in n n b nn n s sho a si gle sketch . Whe egi i g a perspective draw in be an n n neces g which is to made from y projectio drawi gs , it is sary to acquire a thorough knowledge of the form and details

f b The h nn be n n n o . the o ject work , ot erwise , ca ot do e i tellige tly

nd b a rapidly . The perspective sketches for most o jects should be commen ced by drawing in perspective the edges of either — r n b b o . t circumscri ed i scri ed solids , usually square prisms Mos machines and architectural forms are easily treated in this

n b be n way . The pri cipal edges of the o ject should the grouped

n m n n n o n- o f i to three syste s , correspo di g to three parallel edges

O ne be n b n and the prism . of these will represe ted y vertical li es ,

n n A two others must be shown with their proper co verge ce . fter

n b b n n the pri cipal edges of the o ject have ee draw , the other

n n n lines are very eas ily represe ted . Whe deali g with complicated forms it is absolutely necessary to follow some deﬁnite system to obtain results showing reason able accuracy . Even with simple drawings some care shoul d be exercised i n

be n the selecting the poin t of sight. It should take so that details which are considered most importan t will appear in the

n b n o f perspective sketch as plai ly as possi le . A drawi g is little value in which the important parts are crowded so that they are no t n clearly show .

PRACTICAL EXERCISES

“ 1 3 n o f n 9 . b Make a perspective drawi g a scree with four lades , using for the cons truction measuring and vanishing

n poi ts .

1 4 n end nn n 9 . Make a perspective drawi g of the co ectio o f a

n truss as shown i the ﬁgure .

w n o f i n Make a perspective dra i g a s mple woode bridge.

in Draw a cottage perspective . Show in perspective the ﬂight o f steps and the ramp in the

in 183 . ﬁgure Ex .

n b Make a perspective drawi g of a locomotive oiler . Make a perspective drawing of the mil l building shown in

n b n b i the ﬁgure . Show also a tall chim ey ehi d the u ld

In g. ND X I E .

r O te — I n the I nde all ﬁ r s No x gu e refer to p ages ; n o n e to th e n umbers o f l s x rcrses o r r s artrc e e e u e . , , ﬁg

An le etw een a ine and a l ane 68 g b L P , Angle between a Pl ane and a Plane o f Pro j ecti o n 80—84 n le etw een ntersecti n ines 64 66 A g b I g L , , n le o f efl ecti o n 72 A g R , n les fo r afters o f o o fs 74 76 A g R R , ,

A sh - ch ute ead 1 62 165 H , , u iliar l anes 40 46 52 126 A x y P , , , ,

ath - tu b ro lem s 1 30 1 34 B P b , , eltin ro lem s 136 B g P b , ent late Co nnectio n 1 56 15 B P , , 7 e els fo r afters 74 76 B v R , , o at ro l em s 66 70 B P b , ,

o iler- h o u se F u e 1 34 B l , o iler ro lems 142 144 1 58 159 B P b , , , ,

o o t fo r a to e- i e 1 32 B S v p p , rid e ier ro lem 90 B g P P b , rid e ro em s 32 78 80 84 90 B g P bl , , , , , uttress ro lem 1 62 163 B P b , ,

Cam -wheel , 56

Ceilin ro lem 1 58 g P b , Chimne ro lem 134 y P b , Chimne Shado w o n a Roo f 1 0 y , , 7

i rcle o f the Go r e 1 80 C g , Ci rcles in an O li ue l ane ro ecti o n s o f b q P , P j , n 1 Co e , 00

Co ne o f a s 1 30 1 32 134 136 R y , , , , Co nical rch 1 4 6 A , Co nical E a e ro u h O ut et 1 v T g l , 58 Coni cal o z z e 148 N l , Co ni ca Roo f Co nnection 130 l , . 134 Co ni ca oo f ro em 14 l R P bl , 6 Co ni cal o wer o n a oo f 132 T R , Co nn ectin -rod 138 g , Co nto u r in es 98 L , Co nto ured lan Out lanes 98 P by P , Co n o ute u rface 1 18 v l S , Co n o lutes 102 1 14 v , , Co ordinates S stem E lained 1 2 , y xp , Crane en th o f Gu o es fo r 92 , L g y R p , Cu e ro l em 56 1 94 197 b P b , , , Cur e o f ntersectio n 1 26 v I , C inder 100 yl ,

Derri c ro em s 68 94 k P bl , , D es ro lem s 44 54 1 30 k P b , , , De velo ment o f a Co ne 142 p ,

‘ D e elo ment o f a C linder 128 v p y , D e elo m ent o f a lo e Sheet 1 59 v p S p , D e elo ment o f a team D o me 159 v p S , D irectio n o f a s 167 168 R y , , D i rectri ces o f a ar ed urface 1 72 W p S , D i stance etw ee n arallel lanes 56 b P P , , D istance etw een Two ines 92 b L , D i stance fro m a o int to a ine 90 P L , D istance fro m a o int to a l ane 54 P P , ” D istance s r e Meas red 48 t u ) u , , 50

D o me w ith E i ht Sides 1 56 g , D otted ines 14 L , D o rmer indo w ro lems 170 W P b , D o u le Cu r ed u rfaces 100 b v S , D o u l uled u rfaces 176 180 b y R S , , D r D o c ro lem 86 y k P b ,

E a e ro u h O u t et 1 58 v T g l ,

E leme nts o f a u rface 100 126 S , , E le atio n D raw in 38 v g , E lli so id hado w o f 168—170 p , S , E lli tical Cam - w heel ro ectio ns 56 p , P j , “ ” E nd Co nnectio n o f a ru ss 1 98 T , E n ine C li nder ro lem i n Shado w s g y , P b E h au st ead ro lems 148 150 x H P b , ,

Fan ro em 1 38 P bl , ” i ntersectio n 1 2 i I , n i n 1 Fi rst Ge erat o , 76

Mil - u i din 198 l b l g, Mine ro lem 1 56 P b , Mirro r ro lems 44 72 P b ,

e ati e Directio n 12 N g v , otati on 10 12 N , , Nut ro lem 138 P b ,

O li ue Co ne 102 b q , O li ue C linder 102 b q y , O i ue eli co id bl q H , li u e ines to lanes o f ro ection 16 O b q (L ) P P j , ff se o o fo r a Furn ace 1 62 O t B t ,

il -can ro lem 1 58 1 60 O P b , ,

O ne ro ectio n o f a ine in a lane i en to Find the th er 1 P j L P g v , O , 6 ri in 2 O g , r ho ra hic ro ecti o n 1 4 O t g p P j , , “ ” al to e— i e Co nn ectin oot for 132 Ov S v p p , g B ,

arallel ines a lane thro u h 28 P L , P g , arallel i nes ro ectio n s o f 26 28 P L , P j , ,

arallel ines to lanes o f ro ectio n 16 P (L ) P P j , arallel lanes D i stance etw een 56 P P , b , aral e lanes D raw n at a Gi en D istance a 2 P l l P v Ap rt, 6 , 94 attern s fo r an rch to ne 146 P A S ,

er endi cu lar ines to a ane P p (L ) Pl , 54 ers ecti e D raw in 1 188 P p v g, , ers ecti e o f a Cu e 194 197 P p v b , , ers ecti e o f a Ci rc e 1 94 19 P p v l , , 7 ers ecti e D i sto rtio n 1 94 P p v , ers ecti e S etches from Wo rkin D i n s P p v k g raw g , 196 i ctu re lane 1 88 P P ,

i e- ﬁttin ro em s 142 144 152 P p g P bl , , , i e ro lem s 24 46 54 130 132 134 146 148 156 P p P b , , , , , , , , , , itch o f a eli 1 16 120 P H x, , itch o f crew Surfaces 180 P S , lan D rawin 38 P g,

ane D irecto r of a War ed Surface 1 2 Pl p , 7 lanes o f ro ection 2 P P j , lanes 10 P , lane urface 100 P S , lane an ent to a Co ne o r a C linder 108 1 12 P T g y , , lane an ent to a Co n o ute Su rface 1 22 P T g v l , lane an ent to a er o o id o f e o utio n 180 183 P T g Hyp b l R v l , , lane an ent to a S here 1 1 2—1 14 P T g p ,

ane th ro u h a oint er endi cu lar to a i ne 60: Pl g P P p L , ate Co nnection fo r Steel Rafters 1 Pl , 56, 157 oint o f i ht 188 P S g , o rtal o f a rid e 2 P B g , 3 o siti e D irecti on 12 P v , ro ect a ine u o n a ane 56 P j L p Pl , ro ectin ines 4 1 4 22 P j g L , , , ro ectio ns o f ntersectin ines 28 P j I g L , ro ecti ons o f a ine 1 2 16 P j L , , ro ectio ns o f arallel ines 26 28 P j P L , , ro ecti o n s o f a o int 2 1 P j P , , 0 ro ectio n s of o i nts o n a Co ne 104 P j P ,

ro ectio n s o f o ints o n a Co n o l ute u rface 1 2 P j P v S , 0 ro ecti o ns o f o ints o rr a C li nder 104 P j P y ,

ro ecti o n s o f o ints o n a D o u le-cu r ed u rface 104 P j P b v S ,

ro ecti o n s o f o ints o n elico ids o r crew u rfaces 1 82 P j P H S S , ulle ro lem s 96 98 1 36 162 P y P b , , , , ul e s ro ections o f 96—98 P l y , P j

afters o f o o fs n es fo r 74 76 R R , A gl , ,

ailro ad- cut ro em 98 R P bl , i ht Co ne 102 R g , i ht C linder 102 R g y , i ht eli co id 182 185 187 R g H , , , oo f ro lem s 22 38 39 56 74 130 132 134 146 1 56 1 2 R P b , , , , , , , , , , , 6 , 170

Sand- bo x o f a o com oti e 1 56 1 58 L v , , Sco o ro lem 158 1 60 p P b , , - r i e 1 n i h Fo u r lades . e s ect 1 90 93 19 Scree w t B P p v , , , 8

cre S u are—th readed 120 182 185 S w , q , , ,

Screw V- th readed 1 82 185 , , ,

n ra io n 1 Second Ge e t , 76

Sewer ro lem 1 54 P b ,

o r an E lectri c am 160 Shade f L p , h ade ine 1 66 S L , an d h ado w s 1 66 Sh ades S , — a o n a Co umn 168 170 Shadow of a C p l , n o n a lane 1 68 169 Shadow o f a Co e P , , — an E lli so id 1 68 170 Shadow o f p , w o f a h ere 1 70 Shado Sp , ro em s 70 Shado w P bl , — ro em s 56 96 98 Sh aft P bl , , - — a atterns 130- 136 144 145 156 158 165 Sheet met l P , , , , am 130 Sh o rtest Se , ” n ersectio n 12 si I t , cal ro ectio n 2 4 Side Verti P j , , - ved Surface 100 Single cur , S ew rid e ro lem s 32 78 80 84 k B g P b , , , , lo e heet o f a o co m o ti e 1 56 1 59 S p S L v , , here Sh ado w o f 1 70 Sp , , heri cal Hu b fo r a Fan 138 Sp , iral rin ro lem s 1 22 Sp Sp g P b ,

S u are- th readed crew 120 182 185 q S , , , Stair am 1 70 198 R p , , Steam D rum ro lem 142 P b , Steam I i e ro lem s 46 54 p P b , , Ste s ro l em s in h a o w s 170 p , P b S d , Sto e— i eE l o w 1 62 v p p b ,

to e - i e ro lem 132 S v p p P b ,

Stu E n d o f a Co nnecti n - ro d 138 b g ,

u rface o f e o l uti o n 102 S R v ,

ele h o n e ro lem h o rtest i ne 92 T p P b ( S L ) ,

h ree o in ts D etermi ne a l ane 28 T P P ,

o ru s 1 38 T , o w er ro lem 1 46 T P b ,

races o f lanes 10 1 2 T P , ,

s l an s Co irstr ct from r e n es race o f e . u T u T P A gl ,

ran siti o n Co n necti o n 1 64 T ,

rian le ro lem s 56 68 T g P b , , rue n e etw ee n ntersecti n i nes 64 66 T A gl b I g L , , rue D istance etw een T w o o ints 50 T b P ,

rue en th o f a ine 50 T L g L ,

r e iz e o f a lane u rface 62 T u S P S ,

u n nel ro lem 1 56 T P b ,

w i sted u rfaces 100 T S ,

an i shin g o int 194 V P ,

au lt ro l em 1 58 V P b ,

en til atin Fan ro lem 138 V g P b , Verte o f a Co n e 100 x ,

V - th readed crew 182 1 85 S , ,

ar ed urfaces 100 172 W p S , ,

ater- i e ro lem 24 W p p P b ,

i re o o ro lem 1 60 W L p P b , o o den rid e ro l em 198 W B g P b ,

” Y fo r a lo w - i e 144 145 B p p , ,

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sso ci at o n o f tate an d at o n a o o an d air e artm en ts art o r A i S N i l F d D y D p , H f d eet n 1906 8 y o 3 00 M i g , , $ am est o w n eet n . 8 v0 3 00 J M i g , , ’ u s ten s o tes fo r em c a tu en ts . 12 m o 1 50 A N Ch i l S d , ’ as er e h em c a I r ara s n e t o n . B k vill C i l Elem en ts . ( P p i ) ’ B — - ern a do u s m o e ess o w er . i tro c e u o se an d h eo r o f th e e u o se S k l P d N ll l , T y C ll l m 1 2 0 , 2 50 ’ t s I I r an c h m str Ha n tr c t n to o e i . l an d h e an o u . 2 m . 1 o 1 2 5 Bil z d io n g i C y ( l P l , a r r f r mi s r a an o ato et o s o I n o an c e t . d an c L b y M h d g i Ch y ( H ll Bl har d . )

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f n c a r u c ts . Han au sek s cro sco o ec o . . 8v0 5 00 Mi py T h i l P d , ’ m s r m as ins an d ac eo s r an c h e t . 1 2 0 2 00 H k M l d O g i C i y , ’ R Re eren c e a es o n ers o n ac to rs 16m 0 m o r er n s ea . 2 50 H i g dy f T bl (C v i F ) , ’

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n e ts n m a ro u c ts an d atu ra a ters . . 8v0 2 00 , A i l P d , N l W , ’ Lassar- Co hn 5 Applicatio n o f So m e G en era l Reac tio n s to I n vestigatio n s in m . . 2 r an i c h em str . n e 1 o 1 00 O g C i y (Ti gl ) , ’ Lea c h s I n spec tio n an d An alysi s o f Fo o d wi th Spec ial Referenc e to Sta te

8 v0 , 7 50 ’ Lo b s ec tr ch em str o f r an c o m o un s . o ren . . 8vo 3 00 El o i y O g i C p d ( L z , ’ s o tes o n ssa in an d eta u r i c a a o rato r er men ts 8 y o 3 00 Lo dge N A y g M ll g l L b y Exp i , ’ o w s echni ca etho o f O re . 8vo 3 00 L T l M d , 4 ’ o w e 5 a n t fo r tee tru c tu res 1 2 m o $ 1 00 L P i S l S , ’ m 00 . 1 2 0 1 u n e 3 ec n o c em c a n a s s . o n . W L g T h h i l A ly i (C h , ' Mc Ka an d arsen 3 r n c es an d rac t c e Of B u tter m a i n 8 vo 1 50 y L P i ipl P i k g , ’ ‘ MaI re 3 o ern P I m en ts an d t e r e c es 1 2 m o 2 00 M d g h i V hi l , ’ m c a a r r m 1 50 an e s an o o fo r B i o c e o ato . . 1 2 0 M d l H db k h i l L b y , ’ M ar tin s Labo rato r y G u id e to Q u ali tative An alysis with th e Blo w pipe m 0 1 2 , 0 60 ’ f r m d i m aso n s am n at o n o ate . e ca an ac ter o o c a 1 2 o 1 2 5 M Ex i i W (Ch i l B i l g l ) ,

- o n s ere r n c a ro m a an tar tan o n t . Wa ter su pply . (C id d P i ip lly f S i y S dp i )

8 v0 , 4 00 ’ c m c a V 0 at ew so n 5 rst r n e s e . 8 1 00 M h Fi P i ipl Of Ch i l , ’ att ew s 5 a o rato r an u a o f e n an d e t CheririStr 8 v0 3 50 M h L b y M l Dy i g T x ile y , t n Rew r t e t e res . 2 d o t en 8 v0 4 00 T x il Fib Edi i , i , ’ f R c r r r n e . M eye s D ete m in atio n o adi l es i n Ca b o n Co m po u n d s . ( Ti gl h r 1 2 m o H 2 5 T i d , ’ m 0 er s an e ro c ess . 1 2 H 00 Mill Cy id P , I an u a o f ssa n 1 2 m o -00 M l A yi g , ’ r c f m n m an d i ts I n u s r 1 2 m0 50 n e t s o u t o n o u u t a U se . . Mi P d i Al i d i l , ' l h n i c l a c u a t o n s f r u ar r 2 m 0 Mi tte staed t s T ec a o o s . o u r a s . 1 e50 C l l i S g W k (B b ki ) , ’ ter 5 em en tar e t o o o f em str 1 2 m o H 50 Mix El y T x b k Ch i y , ’ f s s r o r an s em en ts o c a e m t “ ’ 1 2 m o w 00 M g El Phy i l Ch i y , h e r f u t o n s an di tsRsu s m u t n e t e o o o t . 1 2 o H 00 O li Of Th y S l i e l , s c a e m str fo r ec tr c a n i n eer s 1 2 m o H 50 Phy i l Ch i y El i l E g , ’ 1 an - su r r O o rse s a c u a t o n s u se in e a ac to r es 1 6m o m o . H 0 M C l l i d C g F i , ’ r s s 0 u r s s to r o f em c a eo e an d aw . 8 v0 h 0 M i Hi y Ch i l Th i L , ’ ~ M u lliken s G en era l M e tho d fo r th e I den tiﬁc atio n o f P u r e O r gan ic Co n po u n ds . 1 l r o n w r r v I . o m u n s a t o en an d en . a e 0 V o . o 8 0 C p d Of C b i h Hyd g Oxyg L g , l s I I I . tr en u s o m o u n . n re rat n . V o . Ni o g o C p d ( P pa io ) l ‘ V . I I I Th m m er a es u f s r o . e o c t a e 8 v0 U 00 C i l Dy f L g , ’ ’ m en f: r O D ri sco ll s o tes o n th e reat t o G o es . 8 vo N 00 N T ld O , ’ ‘— tw a s n ersat n s n em str . ar t n Ra m se 2 m s o o o O e . 1 o l 50 O ld C v i Ch i y P ( y ) , m a rt Tw o . u rn u . 1 2 o M 00 P ( T b ll ) , ’ w n d Stan da c s ein an d ean n o f e t e a r c s 1 2 m N e an . o 00 O g Dy g Cl i g T x il F b i , ’ f s r m o a m er s rac t ca est o o o em t . 1 2 H 00 P l P i l T B k Ch i y , ’

. 1 2 m au s s ca em str in th e er ce o f e c n e . sc er . 0 H P li Phy i l Ch i y S vi M di i (Fi h ) , ’ P-en ﬁ eld s a es o f n era s I n c u n th e U se o f n era s an d ta t s t c s T bl Mi l , l di g Mi l S i i

o f o m es t c ro u c t o n . 8 v0 1 00 D i P d i , ’

h em c a n st tu n . v c tet s a o s an d t e r o t o . e 8 0 5 00 Pi Alk l id h i C i l C i i ( Biddl ) , ’ o o e s Calo riﬁ c o w er o f u e s 8 v0 3 00 P l P F l , ’ resco tt an d n s o w s em en ts o f ater ac ter o o w t ec a Re er P Wi l El W B i l gy , i h Sp i l f

en c e to an tar ater n a s s ” . 1 2 m o 1 50 S i y W A ly i , ’

Re s s G u e to ec e e n “ . 8 v o 2 5 00 i ig id Pi Dy i g , ’ Ri c ar s an d o o m an 3 A i r Water an d Fo o r m a a i tar Stan h d W d , , d f o S n y d

8 v0 . 2 00 ’ Ri c etts an d ll r s No tes o n Assa iii 8 v0 3 00 k Mi e y g , ’ f Ri ea s s n ec t o n an d th e reser at o n o o o . 8v0 4 00 d l Di i f i P v i F d , ew a e an d th e ac ter a u r c at o n o f ew a e 8 v0 00 S g B i l P iﬁ i S g , ’ R s em en tar an u a f o r th e em ca a o rato r 8v0 1 2 5 igg El y M l Ch i l L b y , ’

Ro n e an d en en s an e I n u s tr . Le 8 vo 4 00 bi L gl Cy id d y ( , ’ Ru ddim an s I n co m at ti es in rescr ti o n s 8 vo 2 00 p ibili P ip , Wh s in P h arm ac 2 mo y y ” 1 , 1 00 ’

Ru er 5 em en ts o f etal rah at ew so n . 8 v0 3 00 El M l og py ( M h ) , ’

a in s I n u str a an d rt st c ec n o o o f a n t an d arn s . . 8vo 3 00 S b d i l A i i T h l gy P i V i h , ’

a w s 5 s i c a an d a o ca h em str rn o rf . y o i o o t o . 8 o 2 50 S lk k Phy i l g l P h l gi l C i y ( O d f ) , ’ c m 3 ssen t a s o f o u m etr c n a s s 1 2 mo 1 2 5 S hi pf E i l V l i A ly i ,

R ten ” an u a o f o u m etr c n a s s . t t o n e r t . . 8 v0 5 00 M l V l i A ly i ( Fif h Edi i , w i ) , u a tat e em ca n a s s 8 v0 1 2 5 Q li iv Ch i l A ly i , ’

I n ress . Seamo n 5 M an u al fo r A ssayers an d Ch emi sts . ( P ) ’ m t s ec tu re o tes o n em str fo r en ta tu en ts 8 vo N 8 S i h L N Ch i y D l S d , ’ f r an e u ar an u ac tu rer s r O en c er s an o o o 1 6m o In o . C 8 Sp H db k C S g M f , O f r h m sts o f ee t- su ar r an o o o e 16m o m o . O 8 H db k C i B g , ’ to c r e s Ro c s an d o s 8 y o N 8 S kb idg k S il , ’ O to n e s rac t ca est n o f G as an d G as e ters 8 vo O 8 S P i l T i g M , " ’ O m an s esc ri t e G en era em str . 8 vo O 8 f Till D p iv l Ch i y , em en tar esso n s i n eat 8 y o H 8 El y L H , ’ l rea w e s u a tat e n a s s . a 8y o N 00 T d ll Q li iv A ly i ( H l ) . A uan t tat e n a s s . 8 y o V 00 Q i iv A ly i , 5 ’ T u rn eau re an d Ru ssell s P u blic Water - su ppli e s ’ V a e en ter s s ca em str f r e n n rs n o e . B l D v Phy i l Ch i y B gi ( o tw o o d . ) ’ Ven able s Me tho d s an d D evice s fo r B ac teri a l Trea tmen t o f Sew age ’ r d i e s res w ater I r a an n es s . W d Wh ppl F h Bio lo gy . ( P ) ’ re s ee t- su r u c n R vo P- a a an a tur e a d e n n . V o l . I . . 8 I W B g M f ﬁ i g , N V o l . . 8vo , C

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a s n f th e h m c s f R c . vo o I s an u a o e a n a s o o s . 8 N W hi g M l C i l A ly i k , ' J ea e r s tar o s e s . . 8 vo O W v Mili y Expl iv , ’

e s 5 a o rato r G u e I n u a tat e hem c a n a s s . 8 vo H W ll L b y id Q li iv C i l A ly i , Sho r t C o u rse I n I n o rgan i c Q u ali tative Che m ic al An alysi s fo r E n gin ee rin g i— 1 2 o tu en ts m l S d , ‘ - — f r t m e t c 1 m o e t o o o em c a 2 F T x b k Ch i l A i h i , ’ - O f r n n w a t r . v e s c ro sc o o e . 8 o Whippl Mi py D i ki g . O ’ ‘— n t ro cess 1 2 m o so n s o r a o n I Wil Chl i i P , ‘— 1 2 m o an e ro c e sses . i Cy id P , ’ f s v n to n s c ro sco o e eta e o o . . 8 o H Wi Mi py V g bl F d , ’ mo e an er . ar 2 Z s mo n s o o s an d th e U tram c ro sco e . e 1 w ig dy C ll id l i p ( Al x d ) L g ,

E N NG CIVIL NGI EERI .

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’ ’ a er s n n eers u r e i n I n s tru men ts 1 2 m o 3 00 B k E gi S v y g , ’ s G ra c a o m u t n a r c Bixby phi l C p i g T ble P a pe 1 9 § X24 } I n h es . 2 5 ’ d sm er 3 r c V o l I . e m r an o n es an d a c f r n . en B eed H P i ipl Pr c ti e o Su v e yi g . El tar 8 y o 3 00 y ,

V o l . I I . er u r n 8 vo N 5 0 High S veyi g , ' J u rr 5 n c en t an d o ern n i n eer n an d th e I st mi an an a 8 vo O 50 B A i M d E g i g h C l , ’ m f r n n r o m sto c s e s tro n o o ee s “ . 8 vo M 50 C k Fi ld A y E gi , ’ ‘ o rt e s o w a e ressu re o n ee o u dat o ns 1 2 m o -2 I 5 C h ll All bl P D p F n i , ’ ran a 5 e t o o o n G eo es an d east u ares 8 vo w 00 C d ll T x b k d y L Sq , ’

a s s e at o n an d ta a a es . 8 vo w0 D vi El v i S di T bl , 0 ’ t n r f r m o - o tt 3 n ee n o an . 1 2 t 50 Elli E gi i g L d , rac ca ar r n R r t 2 dti n e te . 1 m t m a a e . eco n n o H P i l F D i g ( S d E i o w i . 50 ’ Fi eb e ger s Treati se o n Civil m00 ’

emer s ho to ra h c e t o s an d I n stru m ts . 8 vo m00 Fl P g p i M h d en , ’ O r s d a n ten an ce . o w e s ew e a e . e n in an 1 z z . 8 vo O 00 F l ll S g ( D ig g M i ) , ’ O re ta s rc tec tu ra n i n eer n . . 8 vo O 50 F i g A hi l E g i g , ’ Go o dhu e 5 M u n ic ipal 12 mo H 50 ’ R c s a s f f r Pr mn ar Est au c an d e e o u an t t s o mat s . . 1 2 In o H 2 5 H h i T bl Q i ie eli i y i e , ’ J

f r n m . a o r 5 e t o o o G eo et c s t o o . 8 vo O 00 H yf d T x b k d i A y , ’

Rea Re eren c a es o n ers o n Fac t rs . m er n 5 e o 16 o m o r . M50 H i g dy f T bl (C v i ) , ’ r A z i m h Ho sm e s u t . . m r 16 o , m o . H 00 ’ o w e Reta n n a sfor Ear th 1 2 mo M2 5 H i i g W ll , ’ ’ s u s m n ts f h e n n eer 5 ran s t an d I es t e o t e e 1 6m o b ds . 2 5 v Adj E gi T i L v l , ’ f ur n r n so n s . . eo r an d ra c t c e o e . a m P 0 o e l 2 o I 0 J h (J B ) Th y P i S v yi g L g , ’ ta t c s e ra c an d G ra c e t o s o n so n s . . 8 vo N 00 J h ( L J ) S i by Alg b i phi M h d , ’ '

i w . Ki n n i c u tt n s o w an d ratt 5 u r ﬁc a t o n o e a e I n re arat o n . , Wi l P P i i S g ( P p i ) ’

a an s esc r t e G eo m e tr . 8 vo u 50 M h D ip iv y , ’

err m an s em en ts o f rec se u r e n an dG eo es . 8 vo w 50 M i El P i S v yi g d y , ’ m a an d ro o s 5 an o o fo r u r e o rs 16m m r w 0 err n o o . 0 M i B k H db k S v y , ’

u en t s an e u r e n . . 8 v o w 50 N g Pl S v yi g , ’ en s ew er o n str u c t o n 8 vo 00 Ogd S C i . w

ew er es n . 1 2 m o w 00 S D ig . ’

P arso n s s s o sa o f u n c a Re u se . . 1 8 vo m00 Di p l M i ip l f , ’

atto n s reat se o n vi n i n eer n . . 8vo h a ea th er q 50 P T i Ci l E g i g . lf l , ’ Ree s o o r a h c a raw in an d 4 to w00 d T p g p i l D g , ’ R ea s ew a e an d th e ac ter a u r c at o n o f ewa e ” 8 vo p 00 id l S g B i l P iﬁ i S g , ’

R e m r a t s n n u n er c u t o n t o n s . o m n an d e s ee e . 8 vo 00 i Sh f i ki g d Difﬁ l C di i (C i g P l , ' e ert an d n 5 o ern to n e - cu tt n an d aso nr 8vo 50 Si b Biggi M d S i g M y , ’ M Mi ll 0 m t s anu a o f o o r a c a raw n . c a . vo w S i h M l T p g phi l D i g ( n 8 5

’ f r u Co fﬁ n s G r a c a o u t o n o a c ro em s . 1 6m m r o o . 2 50 phi l S l i Hyd li P bl , $ ’

at er s n am o m e ters an d th e easu rem en t o f o w er . 1 2 m o 3 00 Fl h Dy , M P , ’

5 r s n i n r n . o w e a te u ee . 8 vo 4 F l ll W pply E g i g , ’ ll s ater - o w er Fri e . 8 v o 5 g W p , ’ u er te s 3 a ter an d u c ea t m . l 2 o 1 F W P bli H l h , 50

a ter tra t o n . 1 2 mo 2 50 W ﬁl i , ’ G an gu i ll e t an d Ku tter s G en eral Fo rm u lafo r th e Uniform Flo w o f Water I n R s r s d t e an n e . er n an d r er an T au tw in e . 8 vo 4 iv O h Ch l ( H i g , 00 ’ s an a er an d Ho w to G e t I t m a en e t . ar e 1 2 o 1 H z Cl W L g , 50 f u c a ter su s tra t o n o e . . 8 vo 3 00 Fil i P bli W ppli , ’ r w rs n d n f rk a e u st s o e a a s o r a ter w o s . 8 vo 2 0 H z lh T T k W , 5 ’ ersc e s 1 1 5 er m en ts o n th e arr i n a ac t of Lar R g, ete eta H h l Exp i C y g C p i y e iv d , M l o n u ts . 8 vo 2 00 C d i , ’ o t an d G ro e r 3 R er sc ar e . 8vo 2 0 H y v iv Di h g . 0 ’ u ar an d Ki ersted 5 ater w o r s an a em en t an d a n ten an c e H bb d W k M g M i . v 8 o , ’ n o n s e e o m en t an d ec tr ca str u t o n o f ater o w e r Ly d D v l p El i l Di ib i W P . v 0 8 o , 0 00 ’ s r - o n s ere r n c a r m M a o n s Wate su pply . (C id d P i ip lly f o a San i ta r y S tan d P o n t . 8 vo ' 00 p i , ’ H rri m an s reat se o n ‘ e . 8 vo O 00 M T i , ’ o to r s rau c s o f Ri ers eirs an dSuces ” . 8 vo N M li Hyd li v , W l i , ’ Ri c ar s 5 a o ra to r o tes o n I n u str a a ter An a ss 8 vo 50 h d L b y N d i l W ly i , ’ c u er s Reser o rs fo r I rr a t o n a ter o w er an d o m e s t c a ter S h yl v i ig i , W p , D i W R s su . ec o n t o n e e an d n ar e . ar e 8 vo m00 pply S d Edi i , vi d E l g d L g , ’ h r f s m an d a tt 3 I m e m en t R ers . o as o o . 4 to c 00 T W p v iv , ’ i - T u rn aur d Ru sse s u c ater su es ” e e an . 8 vo o 00 ll P bli W ppli , ’ b n s n s r f s th E r We r an n 3 e n an d o t u c t o n o am . 5 d a e d . 4 to c 00 g D ig C i D , enl g , Water - Su pply o f th e City o f New Yo rk fro m 1 658 to 189 5 Ho 00 ’ h e s a u e o f ur e ar e 1 2 m o H 00 W ippl V l P L g , ’ am s an d a en s rau c a es 8 vo H 50 Willi H z Hyd li T bl , ' so n s I rr a t o n n i n eer n 8 vo A 00 Wil ig i E g i g , v M 8 o , 50

RI G I RI G MATE ALS OF EN NEE N .

’ B aker s Ro a ds an d Pa vem en ts re a t s n a s n r o n s tru c t T i e o M o y C io n . ’ c s n e ta es u c o r s Bla k U i t d S t P bli W k ” ’

c r u m n u R a I n Pr ss . Blan h a d s Bi t i o s o d s . ( e ) ’ l s n u ac u r o f r u c m I n re r B ei n i n er a t e a e en t . g M f Hyd li C ( P pa ati o n . ) ’ o e s tren t o f a ter a s a n d eo r o f tru c tu re s 8 vo B v y S g h M i l Th y S , ’ rr t c t a n d Res stanc e o f th e ater a s o f n i n ee r n u s as . . 8 v o B El i i y i M i l E g i g , ’

rn e 5 w a . 8 vc By High y , I n sp ec tio n o f th e M a teri a ls andWorkman ship Em plo ye dI n C o n s truc ti o n 1 6m o , ’ h u rc s ec an c s o f n n ee r n 8 y o C h M h i E gi i g , ' c an c s f n n e r n D u B o i s s M e h i o E gi e i g . l e m t c s ta c s s V o . I n a t n e t c . m a 4 to 7 50 Ki i , S i , Ki i S ll , h re ss s i n r m e tru c tu r e s r f V o l . I I . T e t e a t en t o a ter a s an d S F d S , S g h M i l f r s eo r o e u e . SIn a ll 4 to Th y Fl x , ’ c e s em en ts m es an d as ters 8 vo E k l C , Li , Pl , a r u c ts u s i n E n i n I n re r to n e an d o e eerI n . a a n S Cl y P d d g g ( P p tio . ) ’ o w er s r n ar o un a t o n s 8 vo F l O di y F d i , ’ G reen e s tr uc tur a ec an c s 8 vo S l M h i , ’ o e s ea an d Z n c m en ts ar e 1 2 m o H ll y L d i Pig L g , ’ o e an d a s n a s s o f e a n ts o o r P I m en ts an d V a rn I sh es H ll y L dd A ly i Mix d P i , C l g ar e 1 2 mo 2 50 L g , ’ r I n r r u ar s u st re en t es an d Ro a n e s . e a at o n H bb d D P v iv d Bi d ( P p i . ) ’ R e th o s fo r th e em c a n a si s o f o n so n s . . a ec a tee s J h (C M ) pid M d Ch i l A ly Sp i l S l , - G ra te . tee m a i n o s an d . ar e 1 2 mo S l k g All y phi L g , ’

i a s o f o n stru c t o n . r o n so n s . . ater a e 8 vo J h ( J B ) M l C i L g , ' ee s ast I ro n 8 vo K p C , ’

an a s e ec h an cs . 8 vo L z Appli d M i , ’ o we s a n t fo r tee tru ctu res 1 2 m o L P i S l S . , 8 ’ a re s o ern m en ts an d t e r 1 2 m o M i M d Pig h i , ’

Marten s s an o o o n est n ater a s . en n n . 2 o s 8 vo H db k T i g M i l ( H i g ) v l , ’ au rer s ec n c a ec h an c s 8 vo M T h i l M i , ’ err s to n es fo r u n an d ec o rat o n 8 vo M ill S B ildi g D i , ’ err m an s ec an c s o f ater a s 8vo M i M h i M i l , tren th o f ater a s 1 2 m o S g M i l , ’ an u a fo r tee - u sers m etca s tee . 1 2 o M lf S l A M l S l , ’ o rr so n s w a n i n eer n 8 vo M i High y E g i g , ’ atto n s rac t ca reat se o n o un at o n s 8 vo P P i l T i F d i , ’ R ce s o n c r ete o c an u ac ture 8 vo i C Bl k M f , ’ m n ts R c ar so n s o ern s a t a e e . . 8 vo i h d M d A ph l P v , ’ ’ rem a 8 c et o o andRea R r 1 6 r R h e 3 u n o n o e e ce . 6 6m o c . i y B ildi g F P k B k dy fe n , ' ’ ’ C em en t Wo rkers an d Pl asterer s Editio n ( B u ildin g M ec h an ic s Ready R er n c m o e e e . 1 6 m o r f , . f r r n den ts of o n s ru c o n an o o o u e n t t t . m o r . 16 m o . H db k S p i e C i , ’ ’ S to n e an d B ric k M aso n s Edi tio n ( B u ildin g Mechan ic s Rea dy

Re eren c e er es . m m r 16 o o . f S i ) , ’ Pr e r s Ri s s a s : e r cc u rren ce o t e an d ses ” e . 8 vo Cl y Th i O , p i , U , ’ Ries an d L eighto n 3 Hi sto r y o f th e Clay w o rkin g I n dustr y o f the Un i te d ta tes S 8 vo . ’ I n u stri a n d rt st c ec no o o f a n t an d rn s a n s a a . 8 vo S bi d l A i i T h l gy P i V i h , ’ Sm i th s Stren gth o f M aterial 1 2 m o ’ Sn o w s P ri n c i al S eci es o f Woo d ” v p p 8 o , ’ l Spaldi n g s Hy drau i c Ce m en t ” e t o o o n Ro a s an d 1 2 m o T x b k d , ’ a o r an d o m so n s reat se o n o n c re te Pa n an d Re n o rc e 8 vo T yl Th p T i C , l i i f d , ’ r I n re e ar s Th u rsto n 5 M a teri als o f E n gi n ee in g . Th P t No n m eta c ater a s o f n n eer n an d ar t I . eta u r 8 vo P lli M i l E gi i g M ll gy ” . , ar t I I I ro n an d . . 8vo P , I r s rasses Br s an d ar t I I . eat e o n t er o s an d t e r P A T i B , onze O h All y h i o n st tu en ts 8vo C i , ’

m s an d a n ater a s . i lso n s treet a e en t . 8 vo T l S P v P vi g M i l , ’ Trau tw in e s o n c rete a n an d Re n o rc e 1 6m o C , Pl i i f d , ’ Tu rn eaur e an d au rer s r n c es o f Re n o rc e o n c rete o n s tru c t o n M P i ipl i f d C C i .

Re se an d n ar e . eco n t o n . 8 vo S d Edi i , vi d E l g d , ’ ater ur s em en t a o rato r an u a 1 2 m o W b y C L b y M l , ’ ’ h R s an c f r o o s D e . reati se o n t e es t e o a te a s an an en o n W d ( V ) T i M i l , d App dix

th e reser at o n o f m er . 8 vo 2 00 P v i Ti b , ’ Ru s ess o at n s : o rr o s o n an d ec tro s Woo d s ( M . P . ) tl C i g C i El ly i s o f I ro n an d tee 8vo 4 00 S l ,

R I Y GI RI G A LWA EN NEE N .

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u tts 5 n n eer s e o o . 1 6m o m o r B Civil E gi Fi ld b k , . ’ C ran a 5 Ra w a an d th er arth rk 8vo d ll il y O E wo , ran s t o n 1 6m o m o r T i i , . ’ h m u a i 3 fo r ar t w r o t t o s . ro c ett et o s . 8 vo C k M h d E o k C p n , ’ 5 s o r o f h n s an a Ra lro a 1 87 D re dge Hi t y t e P en ylv i i d . ( 9 ) P apei ' sh er 3 a e o f u c ar o ar Fi T bl C bi C db d , ’ ’ ’ w Ra r a n i n rs e dbo o an d lo re rs G G o n 5 o ee ui d . 1 m r e 6 o m o . d i il d E g Fi l k Exp , ’ H u dso n s Tables fo r Calcu latin g th e Cu bic Co n ten ts o f Exca vatio n s an d E m m en t b an k s . y . 8 o , 1 00 ’ ms I n Sur e n Ra ro ad u r e n a I es an d ts 5 ro e i i n d G eo es v Hil P bl v y g, il S v y g d y m 1 6 o , m o r . ’ o to r an d ear s an u a fo r Res en t n i n eers 1 6m o M li B d M l id E g , ’ R r n rs a e s e an u a f o r a o a in ee . 1 6m o m o r N gl Fi ld M l il d E g , . ’

O rr o ck s Ra ro a tru ctur es an d s t m ates . 8 vo il d S E i . , ’ P hi lb ri ck s e an ua fo r 16mo m r o . Fi ld M l , ’

o um es. Raym o n d 3 Railro ad E n gi n ee rin g . 3 v l I r l Ra r a e G eo m etr . n r V o . I . o e a ati il d Fi ld y ( P p o n . ) I m n s o f Ra r a n i n eer n l I e e t o . Vo . . El il d E g i g ’

l . I I I Ra ro a n i n eer s i e o o . In re ara n ti o . Vo . il d E g F ld B k ( P p ) 9 ’ ear es 3 e n i n eer n 1 m o m o r 6 . S l Fi ld E g i g , Ra r o a 1 6m o m o r . il d , ’ r s r sm a rmu lae an dEar th rk a o o o . 8 v o T yl P i id l F wo , ’ Trau tw n e 5 e rac t ce o f a n O u t rc u ar u r s f r Ra r a s Fi ld P i L yi g Ci l C ve o il o d . m o r 1 2 , m o . 2 Metho d o f C alc u latin g th e Cu bic Co n ten ts o f Exc avatio n s an d E m

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m c f R r c . ar e 1 2 m o e 5 co n o s o a ro a o n st u t o n . W bb E i il d C i L g , Ra r m r o a 1 6 o m o . il d , ’ e n to n s c o n o m c eo r of the L c a t o n o f Ra w a s ar e 1 2 m o W lli g E i Th y o i il y L g , ’ m o i so n 8 em en ts o f Ra ro a rac an d . 12 W l El il d T k ,

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’ arr s inematics of 8 vc B K , ’ ar t et s ec h an i raw n t ca . 8vo B l M l D i g , v 8 o , ’ f r o o e s an u a o awi n . 8 vo a er C lidg M l D g , p p , ’ Coo lidge an d Freem an s Elem en ts o f G en e ra l Draftin g fo r M ec han i c al E n gi o n 4 to . Obl g , ’ D ur l e s n e m a t c s o f ac in es 8 vo y Ki i M h , ’ mc 5 I n tro u c t o n to ro ec t e G eo m etr an d i ts lic at o n 8 vo E h d i P j iv y App i , ’ ren c an d I es tereo to m ” 8 vo F h v S y , ’ n a dShad s s e t o o o es an o an d ers ec t e . 8 vo Hill T x b k Sh d w , P p iv , ’ s an c e ec an c a r w am o n 5 a n . 8 vo J i Adv d M h i l D i g , f an r em en ts o ec ca aw n . 8 vo El M h i l D i g , ’ Jo n es s M ac hin e D esign I n em at c s o f ac in er art . 8 vo P Ki i M h y ,

m tren h an d r r f arts . . ar t I I . o r t o o t o n s o 8 vo P F , S g , P p i P , ’ m a an d ar r s ac n e es n 8 vo Ki b ll B M hi D ig , ’ f D v MacCo rd s em en ts o escri t i e G eo m e tr . 8vo El p y , n e ma t c s o r rac t c a ec an sm 8 v o Ki i ; , P i l M h i ,

ec an c a . 4 to M h i l , e o c t a r a m s 8 vo V l i y Di g , ’ Mc Leo d s e sc r t e G eo m etr ar e 1 2 m o D ip iv y L g , ’ a an s escr t e G eo m e tr an d to n e - cu tti n 8vo M h D ip iv y S g ,

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Sc h w amb an d err 5 em en ts o f . 8vo M ill El , ’

m t an d ar s ac n e . 8 vo S i h (A . W . ) M x M hi , ' ll v Mc Mi an . 8 o n u a o f o o ra c a ra . m . a S t s R . i h ( S ) M l T p g phi l D wing ( ) , I tsw o r th 5 em en ts o f ec an c a raw n o n 8 vo T El M h i l D i g Obl g , ’ m arren s ra t n I n s tru m en ts an d era t o n s . 1 2 o W D f i g Op i , em en ts o f esc r t e G eo m e tr a o w s an d ers ec t e 8 vo El D ip iv y , Sh d , P p iv ,

e m en ts o f ac n e o n s tru c t o n an d . 8 vo El M hi C i , e m en ts o f a n e an d o ree h an G eo m e tr c a 1 2 m o El Pl S lid F d i l , G en era ro e m s o f a es an d a o w s 8vo l P bl Sh d Sh d , Man u a l o f El em en tar y Pro bl em s i n th e Lin ear P ersp ec tive o f Fo rm s an d a w l 2 rn o Sh do , an f m en tar ro ec ti o n raw n l 2 m o M u al o Ele y P j D i g ,

n r m s I n em en tar . l 2 m o Pl a e P o bl e El y , G o m etr 8vo ro em s eo rem s an d a m es I n esc r pt e , P bl , Th , Ex pl D i iv e y ’ r an sm ss o n erm an n an d Weisb ac h s Kin em a tic s an d P o w er o f T i i . ( H

Klein . ’ h c ur e i n s . To o r a Wi l o n s ( H . M ) pg p i S v y g ’ escr ti e G eo m etr . Wil so n 5 ( V . T . ) D ip v y Free h an d Le tteri n g - Free h an d Pers pec tive . ’ m tar o u rse I n escri t e G eo metr ar e 8 y o Woo lf s Ele en y C D p iv y L g , 10

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a er s t c unc ti on s . . 8vo 1 50 B k Ellip i F , $ ' r m n s f n a c G r s s e e t o an e t eo m et . och er . 1 2 m o 1 00 B igg El Pl A ly i y ( B ) , ’ u c an a s i o n o me tr n an e an d er ca r . . 8vo 1 00 B h Pl Sph i l T g y , ’ B erle s arm o n c u n c t o n s 8vo 1 00 y y H i F i , ’ r m n f h I an d e s e e ts o t e n ﬁni tesim al a c u u s . 1 2 m o 2 00 Ch l El C l l , C ’ o fﬁ n s ec to r n a s s . 1 2 m o 2 50 V A ly i , ' o m to n s an u a o f o ar t mi c o m u tat o n s 1 2 m o 1 50 C p M l L g i h C p i , ’

c so n s o e e e ra . ar e 1 2 m o 1 50 Di k C ll g Alg b L g , I n tro u c t o n to th e h eo r o f e ra c u a t o n s ar e 1 2 m o 1 2 5 d i T y Alg b i Eq i L g , ’ E m c h s I n tro u c t o n to ro ec t e G eo m e tr an d i ts cat o n 8 vo 2 50 d i P j iv y Appli i , ’ s e s u n c t o n s o f a o m e ar a e 8 vo 1 00 Fi k F i C pl x V i bl , ’ a ste s em en tar n t et c G eo m e tr 8 vo 1 50 H l d El y Sy h i y , em en ts o f G eo m etr 8 vo 1 75 El y , 1 2 m Rat o n a G eo m etr . o l 50 i l y , n t et c ro ec t e G eo m etr 8 vo 1 00 Sy h i P j iv y , ’ ’ e s G rassm an n s ac e n a s s 8 vo 1 00 Hyd Sp A ly i , ’ h h r - lace o ar t m c a es : es t- o c e t s e a er o n so n s . . ee 15 J ( J B ) T p L g i h i T bl V p k iz , p p ,

100, c o es 5 00 pi , o u n te o n ea c ar o ar 8 X10 n c es 2 5 M d h vy db d , i h , 10 c o es 2 00 pi , ' f r a a d I n te a a c u s o h n r e t o n s o e en t n r u . J so n s (W . W . ) Ab idg d Edi i Diff i l g l C l l ar 1 2 m o 1 vo l e . 2 50 L g ,

- u r e r ac n i n artes an Co o r n ates . 1 2 m o 1 00 C v T i g C i di , f eren ti a u a t o n s 8 vo 1 00 Dif l Eq i ,

em en tar rea t se o n f eren t a a c u u s . ar e 1 2 m o l 50 El y T i Dif i l C l l L g ,

' em en tar rea t se o n th e I n te ra a c u u s . . a r e 1 2 m o 1 50 El y T i g l C l l L g ,

eo re t c a ec an c s . o 1 2 m o 3 00 Th i l M h i , eo r o f rro rs an d th e e t o o f eas t u ares 1 2 mo 1 50 Th y E M h d L Sq ,

reat se o n f eren t a a c u u s . . ar e 1 2 m o 3 00 T i Dif i l C l l L g ,

rea t se o n th e I n te ra a c u u s . ar e 1 2 m o 3 00 T i g l C l l L g , D f r n t a u n r reat se o n r n ar an d ar t a e e at o s ” . a e 1 2 mo 3 50 T i O di y P i l if i l Eq i L g , ’ Karapeto ff 5 E n gin e erin g Applic atio n s o f Higher M athe m atic s . I n re a ra ( P p tio n . ) ’

r c m r . a ac e s o so c a ssa o n ro a t e s . u s o tt an d o 1 2 m o 2 00 L pl Phil phi l E y P b bili i (T E y ) , ’ Lu dlo w an d B ass s Elem en ts o f Trigo n o m e try an d Lo gar i thm ic an d O ther a es 8 vo 3 00 T bl ,

r o n o m e tr an d a es u s e s e a ra te . . a c 2 00 T ig y T bl p bli h d p ly E h , ’

u o w s o ar t m c an d r o no m e tr c a es . 8 vo 1 00 L dl L g i h i T ig i T bl , ’ ac ar an e s ec to r n a s s an d u a te rn o n s 8 vo 1 00 M f l V A ly i Q i , ' Mc Mah o n s er o c u n c t o n s 8 vo 1 00 Hyp b li F i , ’ M an n in g s I rr a tio n a l N u mbers an d thei r Re presen ta tio n by S equ en c es an d er es 1 2 m o 1 2 5 S i , a c r te an s e err m an an d Ro er M them ati al M o n o g aphs . Edi d by M ﬁ ld M i b t r c h . o o w a ta o eac 1 S W d d O v , 00 f a a c a u e m t s n e m . N . sto r o rn t e t o . 1 o e Hi y M d M h i , by D vid E g S i h tr G r ru c N . n t t c ro ec t e G eo m e eo e e a s o . 2 e te . Sy h i P j iv y , by g B H l d

m n an s Laen as G o r e . No . ter t 4 . er No . 3 . e D i , by iff d W ld Hyp

m es Mc M o n . N . ar m b o lic un c t o n s a ah o 5 . o n ic un c F i , by J H F ’ N G r . m an n s ac n s s t o n s am . er . o . 6 ass e a i i , by Willi E By ly Sp A ly ,

r No . 7 . ro a a r f w a . e . li t n d eo o rro rs by Ed d W Hyd P b bi y Th y E , w ar ec r s R r . No o 8 . to n a s an d uaterni ns o e t o . . o by b S W d d V A ly i Q , a r a ar n N iff eren i a u atio n s e n e c a e . o . 9 . t by Al x d M f l D l Eq , by f s . 0. Th e u t o n o u a i am o o e o hn so n No . 1 o t o ns Willi W l y J S l i Eq , f m e an s e err m an o . 1 1 . un c t o n s o a o aria e by M ﬁ ld M i . N F i C pl x V bl ,

s . by Tho m as S . Fi ke ’

au rer s ec ni c a ech an i cs . 8vo 4 00 M T" h l M . err m an s e t o o f ea st u ares 8vo 2 M i M h d L Sq , 00 o u t o n o f u at o n s 8vo 1 00 S l i Eq i , ’ Ri ce an d o n so n s i f eren t a an d I n te ra a c u u s i n 2 o s . n o e . J h D f i l g l C l l . v l ar e 1 2 mo 1 50 L g , m en tar reat se n th e D iﬁ re i l a c r e o e n t a u us. a e 1 2 m o 3 00 El y T i C l l L g , ’ m t s sto r o f o ern ath em at c s &v0 1 00 S i h Hi y M d M i , ’ Veblen an d Lenn es s I n tro du c tio n to th e Rea l I n ﬁ ni tesi m al An alysis o f O n e 2 00 1 2 ’ ater u r s es c e an — o o o f at em at c s fo r n i n r W b y V t Po k t H d b k M h i E g ee s . 2 X5 n c es m r o . $ 1 00 % % i h , n ar e t n I n c u n a es m r o o . 1 50 E l g d Edi i , l di g T bl ’ W e s e term n an ts 8 vo 1 00 ld D i , ’ o o s em en ts o f Co -o r n ate G eo m e tr 8 vo 2 00 W d El di y , ’

o o w ar s ro a t an d eo r o f rro rs . . 8vo 1 00 W d d P b bili y Th y E ,

ME C N C E NG NEE N HA I AL I RI G .

RI G I RI G - G I I MATE ALS OF EN NEE N STEAM EN NES AND BO LERS .

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B ac o n s o r e racti ce . 1 2 m o 1 50 F g P , ’ a w n s team ea t n fo r u i n s 1 2 m o 2 50 B ld i S H i g B ldi g , ’ arr s n em a t c s o f ac n er 8 vo 2 50 B Ki i M hi y , ' ar t e tt s ec an ca raw n 8vo 3 00 B l M h i l D i g ,

8vo , 1 50 ’ n i n eer n n h a m an a . u rr s n c en t an d o ern a d t e I st n . 8 vo 3 50 B A i M d E g i g h i C l , ’

ar en ter s er m en ta n i ne er n . 8vo 6 00 C p Exp i l E g i g ,

ea t n an d en t at n u n s . 8vo 4 00 H i g V il i g B ildi g , ’ n i n New e t o n i n ress Clerk s G as an d Oil E g e . ( di i p ’

o m to n s rst esso n s in e ta o r n . 1 2 m o 1 50 C p Fi L M l W ki g , ’ ‘ o m to n an d D e G ro o dt s ee ath e 1 2 m o 1 50 C p Sp d L , ’ o o e s an u a o f raw n 8 vo a er 1 00 C lidg M l D i g , p p , ’ Coo lidge an d Fr ee m an s El em en ts o f G een ral Draftin g f o r M ec han ic a l E n

i n eers . o n 4 to 2 50 g Obl g , ’

C ro m w e s rea ti se o n e ts an d u e s . 1 2 m o 1 50 ll T B l P ll y ,

rea t se o n o o t e G ea r n . 1 2 mo 1 50 T i T h d i g , ’ n e s ac n er attern a n . 1 2 m o 2 00 Di g y M hi y P M ki g , ’ f ac es ” v D ur e s in e m a t c s o n . 8 o 00 l y K i M hi , ’ - F an ers s G ear c u tt n ac n er . ar e 1 2 m o 3 00 l d i g M hi y L g , ’ Fl ath er s n a m o m e ters an d th e easu remen t o f o w er 1 2 mo 3 00 Dy M P , R r n 2 m o o e . 1 2 00 p D ivi g , ’ m G s G as an d u e n a s s fo r n n eers . 1 2 o 1 2 5 ill F l A ly i E gi , - ’ G o ss s o co m o t e ar s . 8 y o 2 00 L iv Sp k , ’

r s u m n ac hi n er I n re arat o n . G een e P pi g M y . ( P p i ) ’ er n s Rea Re eren c e a es o n er s o n ac to rs 1 m m r 6 o o . 2 50 H i g dy f T bl (C v i F ) , ’

o ar t an d s s ee n am o ec tr c ac in er . 8 vo 6 00 H b Elli High Sp d Dy El i M h y , ’ Hu tto n s G as n n e 8 y o 5 00 E gi , ’

am so n s an c e ec an c a raw n . 8vo 2 00 J i Adv d M h i l D i g ,

e m en ts o f ec an ca r aw n . 8 vo 2 50 El M h i l D i g , ’

o n es s G as n n e . 8 vo 4 00 J E gi , M ac hin e D esign m c o f ach n er art I . n e at s . 8 vo 1 50 P Ki i M i y , m tren t an d ro o r t o n s o f arts art I I . o r 8 vo 3 00 P F , S g h , P p i P , ’ ’ c c a n n eer s o c e t- o o m o m r K en t s e an . 1 6 o . 5 00 M h i l E gi P k B k , ’ m ss n err s o w er an d o w er ran s o . 8 vo 2 00 K P P T i i , ’ i m a an d arr s ach n e esi n 8 vo 3 00 K b ll B M i D g , ’ Le vin s G as En gi n e 4 00 ’ eo n ar s a c n e o o o s an d et o s 8vo 00 L d M hi Sh p T l M h d , ’

Re r erat n ac n er . o e a en an d o ren s o ern ean . 8 vo 00 L z M d f ig i g M hi y ( P p , H v , D ) , ’

m . Mac Co rd s n e m at c s o r rac t ca ec an s . 8 vo 5 00 Ki i ; , P i l M h i ,

ec an c a raw n . 4 to 0 M h i l D i g , 0 m s v e o c t a ra . 8 o 1 50 V l i y Di g , ’ f r G ases . Mac Farl an d s tan ar Re u c t o n ac to s o r . 8 vo 1 50 S d d d i F , ’

m so n . a an s I n u s tr a raw n . o 8vo 3 50 M h d i l D i g ( Th p ) , ’ Meh rten s s G as n i n e eo r an d es n ar e 1 2 m o 2 50 E g Th y D ig L g , ’

O er s an o o o f m a o o s . ar e 1 2 m o 3 00 b g H db k S ll T l L g , ’ ars a an d o art s ec tr c ac i n e es n . m a 4 to h a eat er 1 2 50 P h ll H b El i M h D ig S ll , lf l h , ’ P eel e s o m resse A i r an t fo r n es 8 y o 3 00 C p d Pl Mi , ’ oo e s Cal o ri ﬁ c o w er o f u e s 8 vo 3 00 P l P F l , ’ o rter s n i n ee r n Rem n sc en c es 1855 to 1882 . 8 vo 3 00 P E g i g i i , , ’ Rei s o urse in ec an c a D raw m 8 vo 2 00 d C M h i l g , - h n c raw in an d em en tar achi n D e t o o o f ec a a e esi n . 8 vo 3 00 T x b k M i l D g El y M g , 1 3 ’ Ri ch ard s s o m resse A i r 1 2 mo $ 1 50 C p d , ’ Ro i n so n s r n c es of ec an sm 8 vo 3 00 b P i ipl M h i , ’ Sc h w am b an d err s em en ts o f ec an sm 8 vo 3 00 M ill El M h i , ’ r m t . . ) an d a s ac n e es n . 8 vo 3 00 S i h ( A W M x M hi D ig , ’ m t s ress - w o r n o f eta s 8 vo 3 00 S i h ( O P ki g M l , ’ u r r So re l s Carb e tin g an d Co mb u stio n i n Alco ho l E n gi n es . ( Wo o dw a d an d

res to n . ar e 1 2 m o 3 00 P ) L g , ’

to n e s rac t ca es t n o f G as an d G as eters . 8 y o 3 50 S P i l T i g M , ’ to n s n m a as a ac n e an d r m e o to r an d th e aw s o f n er t c u rs e s . Th A i l M hi P i M , L E g i 1 m 2 o , 1 00 d o st r i ch reat se o n r c t o n an o n a n er an d o r . 8 vo 3 00 T i F i i L W k M i y Mill W k , ’ so n s o m e te u to m o e I n s tru c to r 1 6mo 1 50 Till C pl A bil , ’ Ti tsw o r th s e m en ts o f ec an c a raw n o n 8 vo 1 2 5 El M h i l D i g Obl g , ’ arren s e m en ts o f ac n e o n s tru c t o n an d raw n 8 vo 7 50 W El M hi C i D i g , ’ a te r u r s est o c e t an - o o o f at em at c s fo r n n r W b y V P k H d b k M h i E gi ee s . 2 5 n c es r X ? m o . 1 00 % 1 i h , t o n I n c u n a es n a r e m r . . o l 0 E l g d Edi i , l di g T bl 5 ’ Wei sbac h s Ki n em atics an d th e Po w er o f Tran smi ssio n ( H errm an n

e n . 8 vo 5 00 Kl i ) , H r r f ran sm ss o n an d G o ern o rs . e m an n ac n e o . 8 vo 5 00 M hi y T i i v ( , ’

o o s u r n es . 8 vo 2 50: W d T bi ,

NG N MATERIALS OF E INEERI G .

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G reen e s tru c tu ra ec an cs . 8 vo S l M h i , ’ s r H o lle y 3 Le a d and Z in c Pigm en t . L a ge 1 2 m o ' o an d a 5 n a s s o f e a n ts o r men ts an d a s e rn es . H ll y L dd A ly i Mix d P i , C lo Pig , V i h a r e 1 2 m o L g , ’ R h s f r h e em c a n a s s f o n so n s . a e t o o t o ec a J h (C . M ) pid M d Ch i l A ly i Sp i l tee s tee - a n o s an d G ra te a r e 1 2 m o S l , S l M ki g All y phi L g , ’ f n s c n r a s o tru t o . 8 vo o n so n s . . ate o J h ( J B ) M i l C i , ’ e s a s I ro n K e p C t . ’ an a s e n c s L z Appli e d M c ha i . ' a re s o ern m en ts an d t e r e c es 1 2 m o M i M d Pig h i V hi l , ’ M au rer s T ec h i n c al M ec h an ics ’ M erriman s M ec h an i cs o f M ateri als . * Stren th o f a ter a s . 1 2 mo g M i l , ’ e tc a s tee an u a fo r tee - u sers 1 2 m o M lf S l A M l S l , ’ f n n s a i n s I n u s tr a an d r t st c ec n o o o a t a d arn . 8 vo S b d i l A i i T h l gy P i V i h , ’ a r a s f ac n es m m s A . te o 2 o t . 1 S i h ( ( W ) M i l M hi , ' r f t r a m s . t en o a e 1 2 m t . t o S i h ( H E ) S g h M i l , ’ h u rs to n s a ter a s o f n n eer n 3 o s 8 vo T M i l E gi i g v l , , — No n m e ta c ater a s o f n n ee r n . vo ar t I . 8 P lli M i l E gi i g ,

I I I ro n an d tee . . vo ar t . 8 P S l , a r t I I I rea t se o n rasses ro n es an d t er o s an d t e r P . A T i B , B z O h All y h i o n s t tu en ts 8 vo C i , ’

D m f t c a ec an c s . y o o s e . e en ts o n a 8 o W d ( V ) El A ly i l M h i , Trea ti se o n th e Re si stan c e o f M a te ri a l s an d an App en dix o n th e reser at o n o f m er 8 vo P v i Ti b , ’ s Ru s t ess o a t n s : o rro s o n an d ec tro s s o f I ro n an d Wo o d ( M . P . ) l C i g C i El ly i

8 vo ,

-EN E AND E STEAM GIN S BOIL RS .

’ err s em eratu re - en tro a r am B y T p py Di g , ’ w er o f e a u rs Ca rn o t s Refl ec tio n s o n th e Mo tive P o H t . (Th to n ’ Chase s A r t o f P attern M aki n g .

’ i s em en ts f vo 4 00 c e o n a t c a ech an c s . . 8 $ Mi h El A ly i l M i , ' r 00 Ro n so n s n c es o f ec an sm . 8 vo 3 bi P i ipl M h i . , ’ an o rn s ec an c s ro e m s ar e 1 2 m o 1 8 S b M h i P bl . L g , ’ Sch w amb an d err s em en ts o f ec an sm 8 vo 3 8 M ill El M h i , ’ m f o o s e en ts o n a t c a ec an c s : . 8 vo 3 8 W d El A ly i l M h i , rin c es o f em en tar ec an c s 1 2 m o 1 8 P ipl El y M h i ,

E D C M I AL .

’ erha en s h si i c h m str i n r ec u res a an d Abd ld P y o lo g al C e i y Thi ty L t . ( H ll

f rc s s . vo n e r n s u ress o n o u e u o o u an . 1 2 m o B h i g S pp i T b l i ( B ld ) , ’ B o ld u an s I mm un e era 1 2 m o S , ’ s u i I mm u n t G a . . o r et t es n . 8 vo B d S di i y ( y ) , ’ Dav en po rt s S tati stic al M etho d s w i th Spe c ial Referen c e to Bio lo gi ca l Vari a t o n s m m r 16 o o . i , ’

r c s o ec te tu es o n I m m u n t o u an . 8 vo Eh li h C ll d S di i y ( B ld , ’ sc er s s o o o f m en tat o n ar e 1 2 m o Fi h Phy i l gy Ali i L g , ’ an u a o f s c a R ff r rs s r sa d o n s . a e 2 m o d e Fu ac hi t . o n o an 1 M l P y y ( C lli ) L g , ’ - Ham m ars ten s e t o o o n s o o i c a e m s tr . . 8 vo T x b k Phy i l g l Ch i y , ’ ac so n s rec t o n s fo r a o ra to r o r i n s o o i c a em str . . 8 vo J k Di i L b y W k Phy i l g l Ch i y , ' a ar- r c r r s r m o ss o n s a t ca U n a n a s . o en . 1 2 L C h P i l i y A ly i ( L z ) , ’ an e s an - o o f o r th e B i o - em c a a o ra to r 1 2 m o M d l H d b k Ch i l L b y , ’ au c h r c f r 1 2 mo s s a em str i n t e e e o e c n e . sc e . P li Phy i l Ch i y S vi M di i ( Fi h ) , ’ - r 2 m o o sco t s o n s an d en o m s an d t e n t o es . 1 P zzi E T xi V h i A ib di , ’ Ro sto ski s e ru m a 1 2 m o no s s . S Di g i , ’

Ru d dim an s I n co m at t es in resc r t o n s . 8 vo p ibili i P ip i , s i n ar m ac 1 2 m o Why Ph y , ’

ff . v s m s r rn o r . . 8 o a o w s s o o c a an d a t o o c a e t . S lk ki Phy i l gi l P h l gi l Ch i y ( O d ) , ’ a tter ee s u t n es o f u m an m r o o 1 2 m o S l O li H E b y l gy , ’ m t s ec tu re o tes o n e m s tr fo r en ta tu en ts 8 v o S i h L N Ch i y D l S d , ’ e s T h o id e e r a r e 1 2 m o Whippl y p F v L g , ’ o o u s tar i en e fo f f c e rs o f th e n e ar e 1 2 m o W dh ll Mili y Hyg O ﬁ Li L g , 1 2 m o erso n a en e . P l Hygi , ’ o rc ester an d t n so n s m a o s ta s s ta s m en t an d a n ten an ce W A ki S ll H pi l E bli h M i , an d u est o n s fo r o s ta rc tec tu re w t an s fo r a m a S gg i H pi l A hi , i h Pl S ll 1 2 m o 1 2 5 o s i ta . H p l ,

ME G TALLUR Y.

’ B etts s e a R ﬁn i ec tr si s L d e n g b y El o ly . ’ B o llan d s En cyc lo pe dia o f Fo u n din g an d Dic tio n ary o f Fo u n d ry Term s u se d i n th e rac t ce o f 1 2 m o P i , m o I ro n o u n er . 1 2 F d , m u em en t . 1 2 o S ppl , ’ o u as s n tec n ca resse s o n ec hn cal u ec ts 1 2 mo D gl U h i l Add T i S bj , ’ l n r a s an d a s : G o ese s Mi e l M et l A Refer en ce B o o k ” ' I l es s ea - s m e t n 1 2 m o L d l i g , ’ o n so n s Ra etho d s fo r th e em c a n a s s o f S ec I al tee s J h pid M Ch i l A ly i p S l , - m o m i n n . a r e 1 2 tee a o s a d G ra hi te . S l k g All y p L g , ’

r . 8 vo ee s ast I o n . K p C , ’ — l - r ss Ch a t i er s h tem atu re u r m s B o u d o u a d ur e . Le e Hig per M eas e en t . ( B g )

1 2 m o , ’ - 1 2 mo etca s tee . an u a fo r tee u sers . M lf S l A M l S l , ’ m in e t s ro u c t o n o f u min u m an d i ts I n u str a U se . 1 2 o M P d i Al d i l , ’ R f at ew so n 8 vo u er s em en ts o eta o r a . El M ll g phy ( M h , ’ h 1 2 mo m t s ater a s o f ac n es . S i h M i l M i , ’ ate an d to n e s o u n r rac t ce 12 mo T S F d y P i , ’ s r s f i n eri n I n r ee ar ts 8 vo Th ursto h a te a o n e . M i l E g g Th P , r N n - m eta c ater a s o f n i n ee r n see n i n eer n a t I . o P lli M i l E g i g , Civil E g i g,

page 9 . I r n an d ee Part I I . o St l I I I rea ti s on B rasses ro n es an d t er o s an d t ei r art . e P A T , B z , O h All y h ’ r U lk e s M o dern Elec tro lytic Co ppe Reﬁn in g . ’ r r c c West s Am eri c an Fo u n d y P a ti e . ’ M o u lders Text B o o k

MI NERALO GY .

’ I re arat ca em n ts . B askerville s Ch emi l El e ( n P p io n . ) ’ ro w n n s I n tro u c t o n to th e Rarer em en ts 8 vo 1 50 B i g d i El , ’ a f eter m n a t e n era P l - ru s s an u o o . en ﬁ e d . 8 vo 4 00 B h M l D i iv Mi l gy ( , ’ an - o o o f n era s u t er s o c et 1 6m o m o r . 3 00 B l P k H d b k Mi l , ’ ester s ata o u e o f n era s . 8 vo a e r 1 00 Ch C l g Mi l , p p , o t 1 2 5 Cl h , ’ C ran e s G o ld an d Silver 5 00 ’ ’ an a s rst en to an a s New s tem o f n e r a o a r e 8 vo 1 00 D Fi App dix D Sy Mi l gy L g , ’ ’ ec o n en to an a s New ste m o f n era o D an a s S d App dix D Sy Mi l gy . ar e 8v o L g , an u a o f n era o an d e tro r a 1 2 m o 2 00 M l Mi l gy P g phy , n era s an d Ho w to tu e m 1 2 m o 1 50 Mi l S dy Th , stem o f n era o ar e 8 vo a e a t er 1 2 50 Sy Mi l gy L g , h lf l h , — f n era . e t o o o o . 8 vo 4 00 T x b k Mi l gy , ' ’

o u as s n tec n c a ress es o n ec n c a u ec ts . 1 2 m o 1 00 D gl U h i l Add T h i l S bj , ’ a e s n era a es 8 vo 1 2 5 E kl Mi l T bl , ’ n e an d a ro u c ts se i n n n eer n I n r E ck el s Sto Cl y P d U d E gi i g ( P e para tio n . ) ’ G esel s n era s an d e ta s : Re eren c e o o m o m o r o l ﬁ . 3 00 Mi l M l A f B k , ’ G ro th s I n tro u c t o n to em c a r s ta o r a a rs a 1 2 m o 1 25 d i Ch i l C y ll g phy ( M h ll) , ’ s an o o fo r e G eo o sts i 6m m r a es o o . 1 50 H y H db k Fi ld l gi , ’

I n eo u s Ro c s . I ddi n s s . 8 vo 5 00 g g k , Ro c n era s 8 vo 5 00 k Mi l , ’ h sen s eterm n at o n o f Ro c —o rm n n e r a s i n n ec o an n t o n s . 8 vo J D i i k f i g Mi l Thi S i , Wi th Thu mb I n d e x 5 00 ’ M ar tin s L abo rato ry Gui de to Q u alita tive An a lysis w i th th e Blo w 1 50 e 1 2 m o 60 pip , ’ - m eta c n era s : e r c c u r ren c e a n d ses err s No n . 8 vo 4 M ill lli Mi l Th i O U , to n es fo r u n an d ec o rat o n 8 vo 5 00 S B ildi g D i . ’ P enﬁ eld s o tes o n etermin at e n era o an d Rec o r o f n era es ts N D iv Mi l gy d Mi l T . 8 vo a er , p p , a es o f n e ra s I n c u n th e U se o f n era s an d ta t s t c s o f T bl Mi l , l di g Mi l S i i o m est c ro u c t o n 8 vo 1 8 D i P d i , ’ Pi rsso n s Ro c s an d Ro c n era s 1 2 m o 2 8 k k Mi l , ’ R c ar s 5 n o s s o f n era 1 2 m o m r o . 1 8 i h d Sy p i Mi l , ’

R es s a s : e r cc u rren ce ro ert es an d U ses . . 8vo 5 8 i Cl y Th i O , P p i , ’ 3 st r f h - t r Ries an d Leighto n Hi o y o t e Clay w o rkin g I n d u s y o f th e Un ite d

8 vo , 2 8 ’ i llm an s ex t- o o o f I m o rtan t n era s an d Ro c s 8 vo 2 8 T T b k p Mi l k , ’ as n to n s an u a o f th e em c a n a s s o f Ro c s 8 vo 2 8 W hi g M l Ch i l A ly i k ,

M N N I I G .

’ r s in e G as an d o sio n B ea d M es Expl s. ’ r Cran e s G o ld an d Silve . I n dex o f Min in g En gin eer in g Literatu re

h s I n r ess . Min in g M et o d . ( P ) ’ Do u glas s Un tec hn ic al Addresses o n Tec hn ic al Su bj ec ts ’ E i ssl er s Mo dern High Explo sives ’ ‘ G o esel s Min erals an d M e tals A Refer en ce B o o k ’ s an u a o f n n I h l sen g M l Mi i g. ’

I les s ea m e t n . 1 2 m o L d S l i g , ’ P eel e s o m resse A i r an t fo r ines 8vo C p d Pl M , ’ ﬁ lt o n t o n s o rn n Un er cu . an d P l Ri em er s a t n n ee e . 8 vo Sh f Si ki g d Dif C di i (C i g ) , ’ ea er 5 tar o s es 8 vo W v Mili y Expl iv . ’ d r i so n 5 rau c an d P l ac er n n . 2 e t o n . ew r tten 1 2 m o W l Hyd li Mi i g di i i ,

reat se o n rac tica an d h eo ret ca n e en t at o n . . 1 2 mo T i P l T i l Mi V il i . 17 AN A E N E S IT RY SCI C .

sso c at o n o f tate an d at o n a o o an d a r e artm en ts art o r A i i S N i l F d D i y D p , H f d 8 vo ee t n 19 06 . M i g , ,

. 8 vo am esto w n eet n 1 9 07 . J M i g , , ’ B ash o re s u t n es o f rac t c a an tat o n 1 2 m o O li P i l S i i , an ta t o n o f a o u n tr o u se 1 2 m o S i i C y H , an ta t o n o f Rec reat o n am s an d ar s 1 2 m o S i i i C p P k , ’

d a n ten an c e . 8 vo Fo lw ell s ew era e . es n n o n stru c t o n an S g ( D ig i g , C i , M i ) , ater - su n n eer n 8 v o W pply E gi i g , ’ o w er s ew a e o r s n a ses 1 2 m o F l S g W k A ly , ’ u er tes s ater - ﬁ l tr a ti o n o r s 1 2 m o F W W k , m ater an d u c ea t . 1 2 o W P bli H l h , ’ G er a r s G u e to an tar I n s ec t o n s 1 2 m o h d id S i y p i , o ern a t s an d a t o u ses 8vo M d B h B h H , an tat o n o f u c u n s 1 2 m o S i i P bli B ildi g , f e t i n Th e ater u ew era e an d u m n o o rn u s . W S pply , S g , Pl bi g M d Ci y B ldi g

8 vo , ' m a en s ean ater an d Ho w to G et I t . ar e 1 2 o H z Cl W L g ,

- trat o n o f u l c a ter su es . 8 vo Fil i P b i W ppli , ’

Kinni c u t n s o w an d r att s u r c at o n o f ew a e . I n re ara t o n . , Wi l P P iﬁ i S g ( P p i ) ’ Leach s I n s pec tio n an d An a lysi s o f Fo o d w i th Spec ia l Referen c e to Sta te

. 8 vo o n tro . C l , ’ i c a 1 m o m c a an d ac ter o o . 2 aso n s am n at o n o f ater . e M Ex i i W (Ch i l B i l g l) ,

- ater su o n s ere r n c a ro m a an tar tan o n t . W pply . (C id d p i ip lly f S i y S dp i ) v 8 o , ’ err man s em en ts o f an i tar n i n eer n 8 vo M i El S y E g i g , ’ O en s ew er o n stru c t o n 8 vo gd S C i , e w er es n 1 2 mo S D ig , ’ P arso n s s s o sa o f u n c a Re u se 8 vo Di p l M i ip l f , ’ resco tt an d n s o w s em en ts o f a ter ac ter o o w t ec a Re er P Wi l El W B i l gy , i h Sp i l f r 2 m en c e to an tar ate n a s s . 1 o S i y W A ly i , ' r ce s an o o o n an i tat o n 1 2 m o P i H db k S i , ’ Ri ch ard s s o s t o f eann ess 1 2 m o C Cl ,

u in e tar es . m o st o f o o . t 1 2 o C F d A S dy Di i , o st o f n as o e an tar c en c e 1 2m o C Livi g M difi d by S i y S i , f e t r m o s t o e . 1 2 o C Sh l , ’ Ri c ar s an d W i lli a m s s e tar o m u ter 8 v o h d Di y C p , ’ m A ir ater an d r a Ri ch ar s an d o o an s o o o m 8. n tar tan d W d , W , F d f S i y S d o n t 8 vo 2 00 p i , ’ ’ ’ ’ Ri c e s u m er s team - fi tters an d n n ers t o n u n h y Pl b , S , Ti Edi i ( B ildi g R ren c e eri es m ec an c s Rea e e l 6 o m o r . M h i dy, f S ) , ’ Rideal s s n ec t o n and th e reser at o n o f o o 8 vo Di i f i P v i F d , ew a e an d ac ter a P u rI fi c atio n o f ew a e 8 y o S g B i l S g , ’ o er s Ai r an d en t a t o n o f u w a s 1 2 m o S p V il i S b y , ’ Tu rn ea ur e a n d Ru sse s u c a te r - su es 8 vo ll P bli W ppli , ’ en a e s G ar a e rem a to r es i n m er c a 8 vo V bl b g C i A i , e th o an d e c es fo r ac ter a rea tm en t o f ew a e 8 vo M d D vi B i l T S g , ’ r w r I r s n ess . War d an d Whippl e s F e h ate Bio lo gy . ( P ) ’ Whipple s Mic ro sc o py o f D rI n k I n g- w a te r r Typho id F e ve . Va lu e o f P u re Wa ter ’ Win slo w s System atic Re la tio n ship o f the C o c c ac eae

M CE NE IS LLA OUS .

’ E mmon s s G eo lo gi c al G ui de - bo o k o f th e Roc ky M o u n ta in Exc u rsi o n o f th e I tern a t o n a o n ress o f G eo sts 4- n i l C g lo gi Lar ge 8 vo . 8 ’ “ h n s u r r e t se o n t e n s . erre o p a a . 8 vo a 8 F l P l T i Wi d , ’ t era s o sto n ac n st 18m o H 8 Fi zg ld B M hi i , ’ G a n s s c a r f h r n ett tat t st ac t o t e o . m . 24 o 8 S i i l Ab W ld , ’ ) ai n s s m e ca Ra m en e r n w a an a e t . 1 2 m o K 8 H A i il y M g , ’ 1 Han ausek s The cro sc o o f c a r c s ec hn o u t . n to n 8vo 8 Mi py T i l P d (Wi ) , 0 18