Appendix 1 Letter from Descartes to Desargues 1 (19 June 1639) Sir, The openness I have observed in your temperament, and my obligations to you, invite me to write to you freely what I can conjecture of the Treatise on Conic Sections, of which the R[everend] F[ather] M[ersenne] sent me the Draft. 2 You may have two designs, which are very good and very praiseworthy, but which do not both require the same course of action. One is to write for the learned, and to instruct them about some new properties of conics with which they are not yet familiar; the other is to write for people who are interested but not learned, and make this subject, which until now has been understood by very few people, but which is nevertheless very useful for Perspective, Architecture etc., accessible to the common people and easily understood by anyone who studies it from your book. If you have the first of these designs, it does not seem to me that you have any need to use new terms: for the learned, being already accustomed to the terms used by Apollonius, will not easily exchange them for others, even better ones, and thus your terms will only have the effect of making your proofs more difficult for them and discourage them from reading them. If you have the second design, your terms, being French, and showing wit and elegance in their invention, will certainly be better received than those of the Ancients by people who have no preconceived ideas; and they might even serve to attract some people to read your work, as they read works on Heraldry, Hunting, Architecture etc., without any wish to become hunters or architects but only to learn to talk about them correctly. But, if this is your intention, you must steel yourself to writing a thick book, and in it explain everything so fully, so clearly and so distinctly that these gentlemen, who cannot study a book without yawning and cannot exert their imagination to understand a proposition of Geometry, nor tum the page to look at the letters on a figure, will not find anything in your discourse which seems to them to be less easy of understanding than the Letter from Descartes to Desargues (19 June 1639) 177 description of an enchanted palace in a novel. And, to this end, it seems to me that, to make your proofs less heavy, it would not be out of the question to employ the terminology and style of calculation and of Arithmetic, as I did in my Geometry;3 for there are many more people who know what multiplication is than there are who know about compounding ratios, etc. Concerning your treatment of parallel lines as meeting at a butt4 at infinite distance, so as to include them in the same category as lines which meet at a point, this is very good, provided you use it, as I am sure you do, rather as an aid to understanding what is difficult to see in one of the types, by comparing it with the other, where it is clear, and not conversely. I have nothing to add on what you have written about the centre of gravity of a sphere: for I have already expressed my opinion sufficiently fully to the R[everend] F[ather] M[ersenne] and your comment at the end of your corrections shows that you have understood what I said. But I ask your pardon if! have allowed myself to be carried away by my enthusiasm in recounting my thoughts so freely, and I beg you to believe me s, ... Appendix 2 Letter from Beaugrand to Desargues 1 (25 July 1639) Sir, I am astonished that you have not received solutions to your problems from those to whom you addressed yourself before, since I find nothing in them that cannot easily be constructed by the propositions of Apollonius. It must be that they are not familiar with the work of this author since they have assured you that they did not see how they could use it; I want to show you how to do so by using the figures of [proposition] 54 [of book] 3, so that you may be left in no doubt. 2 Let there be three points A, H, C, two of which, that is A, C, you suppose to lie on the straight lines AD, CD, whose position is given; you wish to construct a conic section which passes through the point H and touches the straight lines AD, DC at the points A, C. Divide AC into two equal parts at the point E, and draw the line DE, which will be the diameter of the conic that is to be constructed. Draw the straight lines AF, CG, the first parallel to DC, the other to AD, and also the lines AH, H C which meet these lines at the points F, G. Then find on the infinite straight line DE a point or two such as B, such that the ratio of the square of BF to the square of DB is equal to the ratio of the rectangle AF, CG to the rectangle 4AD, DC. Now, if these four rectangles are equal you will only find one point B between D and E which defines, in relation to these points, two lines in the proportion I have prescribed; in this case the [conic] section you require will be a parabola, which you can construct by proposition 20 [of book] 1. 3 But you will find two such points B, K if the rectangles are unequal, and if rectangle AF, CG is greater than rectangle 4AD, CD the conic section will be a hyperbola, and an ellipse if it [the first rectangle] is smaller. And since you know the size and position of the chord5 BK and one of the ordinates, AE, you can describe the ellipse, by proposition 21 [of book] 1. The other problem was to describe a conic passing through four points so arranged that the infinite straight line drawn through two of these points Letter from Beaugrand to Desargues (25 July 1639) 179 passes through the point of intersection of the straight lines which touch the conic at the other two points. Let the four given points be Q, A, H, C and let the line QH pass through the point of intersection of the lines which will touch the required conic at the points A, C. Construct the points Rand D on the line QH such that the ratio of QR to RH is equal to the ratio of QD to DH; the point D is the point of intersection of the tangents mentioned above.4 Having drawn the tangents, it only remains to describe a conic section which passes through the point H and touches the straight lines AD, DC at the points A, C, [which can be done] in the manner I have explained above. However, since, as you see, the second problem reduces to the first, I shall add a second solution to the first problem, a solution which is short and not very difficult. Having drawn the straight lines AF, CG, CH, AH as above, take on the line AF any point Z, and on the line CG cut off CY, a length such that its ratio to CG is the same as that of FA to AZ; joining the lines AY, CZ [i.e. producing (hem until they intersect] their point of intersection, P, will lie on the required conic section, which you can describe by finding as many more such points as you consider appropriate. I could easily give you other solutions to the same problems, either taken from Apollonius or from my own writings on conics, and I could equally render them more general by making certain changes, but I have solved them in the form in which you proposed them, in order to give you greater satisfaction. I am Sir. [sic] Proof of the second construction of the first problem: ae" {eb" bd"{af ' eg 4ad, de{eb" bd" af, eg 4a"d, de ae" aZ,ey 4ad, de eb" bd" af, eg = az, ey af eg {4ad, de ae" eb" db" af eg 4ad de Your very humble servant De Beaugrand Paris 25 July 1639 Appendix 3 Pascal's Essay on Conics (1640) The original title of Pascal's work is Essay pour les Coniques. ESSA Y ON CONICS By B.P. First Definition When several straight lines meet at one point, or are all parallel to one another, all these lines are said to belong to the same order or ordinance [ordonnance, the word used by Desargues], and the set of lines is called an order of lines or an ordinance of lines. Definition II By the term conic section we mean the circumference of the Circle, the Ellipse, the Hyperbola, the Parabola and the Pair of Straight lines, since a Cone, cut parallel to its base, or through its vertex [sommet] or in the three other ways which give the Ellipse, the Hyperbola and the parabola, produces in the Conical surface either the circumference of a circle, or a pair of straight lines, or the Ellipse, or the Hyperbola, or the parabola. Definition III By the word straight [droite = straight], used alone, we mean straight line [ligne droite]. Lemma I [Figure A3.1] -If in the plane M, S, Q we have two lines M K, M V through the point M and two lines SK, SV through the point S, and K is the point of intersection of the straight lines MK, SK and A the point of intersection of the lines M K, S Vi and p.