Development of trigonometry Medieval times Serge G. Kruk/Laszl´ o´ Liptak´
Oakland University
Development of trigonometry Medieval times – p.1/27 Indian Trigonometry
• Work based on Hipparchus, not Ptolemy
• Tables of “sines” (half chords of twice the angle)
• Still depends on radius 3◦ • Smallest angle in table is 3 4 . Why?
R sine( α )
α
Development of trigonometry Medieval times – p.2/27 Indian Trigonometry
3◦ • Increment in table is h = 3 4
• The “first” sine is
3◦ 3◦ s = sine 3 = 3438 sin 3 = 225 1 4 · 4 1◦ • Other sines s2 = sine(72 ), . . . Sine differences D = s s • 1 2 − 1 • Second-order approximation techniques
sine(xi + θ) = θ θ2 sine(x ) + (D +D ) (D D ) i 2h i i+1 − 2h2 i − i+1 Development of trigonometry Medieval times – p.3/27 Etymology of “Sine” How words get invented!
• Sanskrit jya-ardha (Half-chord)
• Abbreviated as jya or jiva
• Translated phonetically into arabic jiba
• Written (without vowels) as jb
• Mis-interpreted later as jaib (bosom or breast)
• Translated into latin as sinus (think sinuous)
• Into English as sine
Development of trigonometry Medieval times – p.4/27 Arabic Trigonometry
• Based on Ptolemy
• Used both crd and sine, eventually only sine
• The “sine of the complement” (clearly cosine)
• No negative numbers; only for arcs up to 90◦
• For larger arcs, the “versine”: (According to Katz)
versine α = R + R sine(α 90◦) −
Development of trigonometry Medieval times – p.5/27 Other trig “functions” Al-B¯ırun¯ ¯ı: Exhaustive Treatise on Shadows
• The shadow of a gnomon (cotangent)
• The hypothenuse of the shadow (cosecant)
• The reverse shadow (tangent)
• The hypothenuse of the reverse shadow (secant) sun sun
reverse shadow
shadow
Development of trigonometry Medieval times – p.6/27 Height and distance via sine Al-Qab¯ıs¯ı:
• Given α1, α2, d
• (angles via astrolabe)
• find x, y using only sines A
y
E
α1 α2
C d D B x
Development of trigonometry Medieval times – p.7/27 Height and distance via sine
d sin α y = 2 sin(90◦ α1) sin α2 sin(90◦ α2) − − − sin α1 y sin(90◦ α ) x = − 1 sin α1
Development of trigonometry Medieval times – p.8/27 Arabic Trigonometry
• Many trig identities
• “Formal” development
• Both plane and spherical trig
Development of trigonometry Medieval times – p.9/27 Regiomontanus 1436–1476
• Real name Johannes Müller
• Translated Ptolemy
• Re-did Ptolemy from basic axioms (à la Euclid)
• Plane and spherical trig
• Many results based on arabic texts
Development of trigonometry Medieval times – p.10/27 Later works
• Tables of all trig functions n • Based on larger and larger radii (10 )
• . . . for accuracy
• . . . no good notation for decimals
Development of trigonometry Medieval times – p.11/27 Names for trig functions
• Regiomontanus 1436–1476: On Triangles of Every Kind
• Sinus complementi
• Joachim Rheticus 1514–1574 • Trig functions from triangles, no circles!
• Thomas Fincke 1561–1656 • Invented words “tangent” and “secant”
• Bartholomeo Pitiscus 1561–1613 • Invents the word “Trigonometry” Everything still depends on a particular radius
Development of trigonometry Medieval times – p.12/27 Pitiscus’s front page
Development of trigonometry Medieval times – p.13/27 From
• A Gentle History for Teachers and Others
• by Berlinghoff and Gouvêa
Development of trigonometry Medieval times – p.14/27 Versine
• Isaac Newton’s Principia 1713
• Charles Hutton A course in Mathematics 1801 • Sine: The line drawn from one extremity of an arc to the diameter, perpendicular to the diameter.
• Versed sine: Part of the diameter intercepted between an arc and its sine.
R sine( α )
α
versine( α )
Development of trigonometry Medieval times – p.15/27 Why tangent? Why secant? Similar triangles (Circle of radius R = 1) a c tan α = = and c is tangent b R R d sec α = = and d is secant b R
d
R c a
α
b R
Development of trigonometry Medieval times – p.16/27 Napier 1614
• Our notation for decimals
• Logs to simplify trig calculations
• (Replacing multiplication of sines by addition)
Development of trigonometry Medieval times – p.17/27 Leibniz (1693) and sine
• Consider a unit circle
• Let x = sin y
dt
dy dx
1 y x
Development of trigonometry Medieval times – p.18/27 x = sin y
• By similar triangles (because dy is tangent) dt x = dx √1 x2 − • By right angle triangle dx2 + dt2 = dy2
• Isolate dt from second, sub into first, square (1 x2)(dy2 dx2) = x2dx2 − − dy2 = x2dy2 + dx2
Development of trigonometry Medieval times – p.19/27 dy2 = x2dy2 + dx2
• Differential, assume dy constant 0 = d(x2dy2 + dx2) = 2 d(x dy) x dy + 2 d(dx) dx 2 2 = 2x dy(dx dy + x d y) + 2 dx dx = 2 dx(d2x + x dy2)
• Therefore d2x = x dy2 −
Development of trigonometry Medieval times – p.20/27 2 d x = x dy2 − • Since sin is an odd function
• and since sin 0 = 0
• Liebniz assumes a power series x = y + cy3 + ey5 + fy7 + . . .
• Differentiate twice d2x = 3 2cy + 5 4ey3 + 7 6fy5 + . . . dy2 · · ·
• Equate term by term both series
Development of trigonometry Medieval times – p.21/27 x = sin y
• Finally 1 1 x = sin y = y y3 + y5 . . . − 3! 5! −
• An expression for sin free of the circle
Development of trigonometry Medieval times – p.22/27 Euler 1730
• Power expansion of exponentials and log
• Power expansion of sine and cosines
• States matter-of-factly a unit circle
• And relations between all of them
• Trig function really as “functions”
• (For Euler, that means analytic)
Development of trigonometry Medieval times – p.23/27 Euler: Unification
iθ iθ e = cos θ + i sin θ e− = cos θ i sin θ − iθ iθ iθ iθ e e− e + e− sin θ = − cos θ = 2i 2 θ θ θ θ e e− e + e− sinh θ = − cosh θ = 2 2 Name due to Lambert : “Hyperbolic trig functions”
Development of trigonometry Medieval times – p.24/27 Why Hyperbolic ?
cosh(a) x^2−y^2 = 1
sinh(a)
1
Development of trigonometry Medieval times – p.25/27 Hyperbolic trig
sinh θ tanh θ = cosh θ cosh θ coth θ = sinh θ 1 sec hθ = cosh θ 1 csc hθ = sinh θ
Verify cosh2θ sinh2θ = 1 − Development of trigonometry Medieval times – p.26/27 Trig: the big picture
• Original concept: chords (depends on radius)
• Use: Solving triangles (plane and spherical)
• Middle-age “sine”: half-chord of double angle
• Liebniz and Euler: Unit circle
• Euler: trig as proper functions
• One Family: exponentials, logs, trig, hyperbolics
Discussion: What is a definition for a mathematician?
Development of trigonometry Medieval times – p.27/27