The ancient Greeks present: Rational Trigonometry

The ancien t Greeks presen t: Rational T rigonometry N J Wildb erger Sc ho ol o f Mathematics and Statistics UNSW Sydney 2052 Australia w ebpages: h ttp://web.m aths.unsw.edu/˜norman/ Octob er 22, 2018 Abstract Pythagoras’ theorem, the area of a triangl e as one half the base time s the height, and Heron’s formula are amongst the most important and useful results of ancien t Greek geometry . Here we lo ok at all three in a new and impro ved ligh t, using quadr anc e not distanc e . This leads to a simpler and more elegan t trigonometry , in whic h angle is replaced by spr e ad , and which extends to arbitrary fields and more general qu adratic forms. Three ancien t Greek theorems There a re three classical theore ms a bo ut triang le s that every mathematics stu- dent tr a ditionally meets. T o state these, c o nsider a triangle A 1 A 2 A 3 with side lengths d 1 ≡ | A 2 , A 3 | , d 2 ≡ | A 1 , A 3 | and d 3 ≡ | A 1 , A 2 | . Pythagoras’ theorem The triangle A 1 A 2 A 3 has a right angle at A 3 pr e cisely when d 2 1 + d 2 2 = d 3 3 . Area of triangle The ar e a of a triangle is one half the lengt h of the b ase times the height. Heron’s formula If s ≡ ( d 1 + d 2 + d 3 ) / 2 is t he semi-p erimeter of a t riangle, then its ar e a is area = p s ( s − d 1 ) ( s − d 2 ) ( s − d 3 ) . In this paper we will reca st all three in simpler and mor e gener al forms . As a reward, w e find that r ational trigonometry fa lls into our la ps, essentially for free. Our r eformulation works ov er a general field (not of characteristic tw o), in arbitrar y dimensions, and even with an ar bitr ary qua dr atic for m—see [2 ], [3] and [4]. 1 Pythagoras’ theorem Euclid and other ancient Greeks reg arded ar e a , not distanc e , as the fundamen- tal q uant ity in pla nar ge o metry . Indeed they work e d with a straightedge and compass in their co ns tructions, not a ruler and pro tr actor. A line s egment was measured by constructing a squar e on it, and determining the are a of that square. Two line segments were considere d equal if they were c ongruent , but this was indepe ndent of a direct notion of distance measurement. Area is an affine c onc ept : more pre cisely pr op ortions b et ween a r eas are maintained by linear transformations. Even with a different metrical ge ometry , for example a relativistic ge o metry in whic h x 2 − y 2 plays the role o f x 2 + y 2 , the notion of signed area defined by a determinant applies. In other words, in planar geometry a r ea is a basic notio n and may be consider ed prior to any theory of linear measurement. T o Euclid, P ythagoras ’ theor e m is a relation abo ut the area s of square s built on ea ch of the sides o f a right tria ngle. This insight has largely b een lost in the mo dern formulation, but with a sheet of gr aph pape r it is still a n attractive w ay to in tro duce studen ts to the sub j e ct, as the area of many simple figures can b e computed by sub dividing, transla ting and co unting cells. A A 1 2 A 3 25 5 20 Figure 1: Pythago r as’ theorem using a reas The squar es on the sides o f triangle A 1 A 2 A 3 shown in Figure 1 have areas 5 , 20 a nd 25 . The lar gest squar e for exa mple can b e seen as four triang les which can b e rearra nged to g et tw o 3 × 4 rectangles, together with a 1 × 1 squa re, for a total area of 25 . So from this p oint of view P ythagoras ’ theorem is a result which can b e established by c ounting , a nd the us e of irrationa l n umber s to describ e leng ths is not necessa ry . This applies to an y right triangle with rational co ordinates. 2 F ollowing the Greek terminology of ‘quadrature’, we define the quadrance Q of a line segment to b e the area of the squa re constructed on it. Py thagoras ’ theorem a llows us to a s sert that if A 1 ≡ [ x 1 , y 1 ] and A 2 ≡ [ x 2 , y 2 ] , then the quadrance betw een A 1 and A 2 is Q ( A 1 , A 2 ) = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 . So for example the quadrance betw een the p oints [0 , 0] and [1 , 2] is Q = 5 . The usual distance b etw een the points is the ‘square r o o t’ of the qua dr ance and requires a prior theor y of irr ational num b ers. Clearly the irrational num b er √ 5 ≈ 2 . 23 6 067 9 77 . . . is a far more sophisticated and complicated ob ject than the natural num b er 5 . In statistics, v a riance is mor e natura l than standar d deviation. In quantum mechanics, wa ve functions are more basic than proba bilit y amplitudes. In har- monic analysis , L 2 is more pleasant than L 1 . In geo metry , quadr anc e is mor e fundamental t han distanc e . F or a triangle A 1 A 2 A 3 we define the quadrance s Q 1 = Q ( A 2 , A 3 ), Q 2 = Q ( A 1 , A 3 ) and Q 3 = Q ( A 1 , A 2 ). Here then is Pythagor as’ t he or em as the Gr e eks viewe d it —a nd it ex tends to ar bitrary fields, to many dimensions, and even with genera l quadra tic for ms. Theorem 1 (Pythagoras) The lines A 1 A 3 and A 2 A 3 of the triangle A 1 A 2 A 3 ar e p erp endicular pr e cisely when Q 1 + Q 2 = Q 3 . Area of a triangle The ar e a of a triangle is one-half t he b ase t imes the height . Let’s see how we might remove some of the irrationalities that o ccur with this ancient formula by loo king at a n example. T he area of the triang le A 1 A 2 A 3 in Figure 2 is one A A F 1 2 A A 3 4 Figure 2: A triangle and an asso cia ted para lle lo gram half of the area of the asso ciated parallelogr am A 1 A 2 A 3 A 4 . 3 The latter area may be calculated b y removing from the circumscr ibed 12 × 8 rectangle four tria ng les, whic h can b e comb ined to form t wo rectangles, one 5 × 3 and the other 7 × 5 . The area of A 1 A 2 A 3 is th us 2 3 . T o apply the one-half base times height r ule, the base A 1 A 2 by Pytha goras has length d 3 = | A 1 , A 2 | = p 7 2 + 5 2 = √ 74 ≈ 8 . 6 02 325 267 04 . . . . T o find the length h of the altitude A 3 F , set the o rigin to b e at A 1 , then the line A 1 A 2 has Cartes ian e quation 5 x − 7 y = 0 while A 3 = [2 , 8] . A well-kno wn result fro m co o rdinate geometry then states that the distance h = | A 3 , F | from A 3 to the line A 1 A 2 is h = | 5 × 2 − 7 × 8 | √ 5 2 + 7 2 = 46 √ 74 ≈ 5 . 34 7 391 3 82 22 . . . . If an engineer doing this calculation works with the sur d forms of b oth ex pr es- sions, she will notice that the tw o o ccurr e nc e s of √ 74 conv eniently ca ncel when she tak es one half the pro duct o f d 3 and h , giving a n a rea of 23 . Howev er if she works immediately with the de c ima l forms, she may be surprised that her calculator gives area ≈ 8 . 602 3 25 267 04 . . . × 5 . 34 7 391 3 8 2 22 . . . 2 ≈ 23 . 0 00 000 000 01 . The us ua l for m ula forces us to descend to the level o f irr ational num b ers and square ro ots, even when the even tual answer is a natural num b er, and this int r o duces unnecessar y a pproximations a nd inaccura cies in to the sub ject. It is not hard to see how the use of quadrance allows us to reformulate the r esult. Theorem 2 (T riangle area) The squar e of t he ar e a of a triangle is one-quarter the qu adr anc e of the b ase times the quadr anc e of the c orr esp onding altitude. As a formula, this would b e area 2 = Q × H 4 where Q is the qua drance of the ba se and H is the quadrance of the altitude to that base. Heron’s or A rc himedes’ T heorem The same triangle A 1 A 2 A 3 of the previous section has side lengths d 1 = √ 34 d 2 = √ 68 d 3 = √ 74 . The semi-p erimeter s, defined to b e one half of the sum of the side lengths, is then s = √ 34 + √ 68 + √ 74 2 ≈ 11 . 3 39 744 206 6 . . . . 4 Using the usual Heron’s formula, a computation with the ca lculator shows that area = r s  s − √ 34   s − √ 68   s − √ 74  ≈ 23 . 0 00 000 . Again w e have a formula inv olving square ro ots in which there app ear s to b e a surprising in teg ral o utcome. Let’s now give another form of Her on’s formula, with a new na me. Arab so urces sugg est that Archimedes knew Heron’s fo r m ula earlier, and the greatest mathematician of all time deser ves credit for more than he currently gets. Theorem 3 (Arc hime des) The ar e a of a triangle A 1 A 2 A 3 with quadr anc es Q 1 , Q 2 and Q 3 is given by 16 area 2 = ( Q 1 + Q 2 + Q 3 ) 2 − 2  Q 2 1 + Q 2 2 + Q 2 3  . In our example the triang le has quadra nces 34 , 68 and 7 4 , each obtained by Pythagor as’ theorem. So Archimedes’ theor em sta tes that 16 a rea 2 = (34 + 68 + 74) 2 − 2  34 2 + 68 2 + 74 2  = 846 4 and this gives an area of 23 . In rational trigonometry , the quantit y A = ( Q 1 + Q 2 + Q 3 ) 2 − 2  Q 2 1 + Q 2 2 + Q 2 3  is the quadrea of the triangle, and turns out to be the single most impo rtant nu m ber a sso ciated to a triangle. Note that A = 4 Q 1 Q 2 − ( Q 1 + Q 2 − Q 3 ) 2 = −         0 Q 1 Q 2 1 Q 1 0 Q 3 1 Q 2 Q 3 0 1 1 1 1 0         . It is instructive to see how to g o from Heron’s formula to Archimedes’ the- orem. In terms of the side lengths d 1 , d 2 and d 3 : 16 a rea 2 = ( d 1 + d 2 + d 3 ) ( d 1 + d 2 − d 3 ) ( − d 1 + d 2 + d 3 ) ( d 1 − d 2 + d 3 ) =  ( d 1 + d 2 ) 2 − d 2 3   d 2 3 − ( d 1 − d 2 ) 2  =  2 d 1 d 2 +  d 2 1 + d 2 2 − d 2 3   2 d 1 d 2 −  d 2 1 + d 2 2 − d 2 3  = 4 d 2 1 d 2 2 −  d 2 1 + d 2 2 − d 2 3  2 = 4 Q 1 Q 2 − ( Q 1 + Q 2 − Q 3 ) 2 = ( Q 1 + Q 2 + Q 3 ) 2 − 2  Q 2 1 + Q 2 2 + Q 2 3  . Archimedes’ theorem implies another for m ula of consider able imp ortance. 5 Theorem 4 (T riple quad form ul a) The t hr e e p oints A 1 , A 2 and A 3 ar e c ol line ar pr e cisely when ( Q 1 + Q 2 + Q 3 ) 2 = 2  Q 2 1 + Q 2 2 + Q 2 3  . The pro of is of course immediate, a s collinear it y is equiv alent to the area o f the triangle being zero. Spread b et w een lines An angle is the r atio of a cir cular distanc e to a line ar distanc e, and this is a c omplicated concept. T o define an ang le pr ope r ly you r e qu ir e c alculus , an impo rtant p oint es sent ially understo o d b y Arc himedes. V agueness a bo ut angles , and the accompanying ambiguities in the definition of the circula r functions cos θ, sin θ and tan θ weaken most calculus texts. There is a rea son that classical trigonometry is painful to studen ts—it is b ase d on the wr ong notions . As a result, mathematics teachers are forced to contin ually r ely on 90 − 45 − 45 and 90 − 60 − 30 triangle s for examples and test questions, which makes the sub ject very na rrow a nd rep etitiv e . Rational trigonometry , developed in [2], see also [4], shows how to simplify and enrich the sub ject, leading to g reater accurac y and quick e r computatio ns. W e want to show that the basic r esults of this new theor y fol low natur al ly fr om the ab ove pr esentation of Pythagor as’ t he or em, the T riangle ar e a t he or em, and Ar chime des’ the or em . The key innov ation is to replace angle with a completely a lgebraic concept. The sepa ration b etw een lines l 1 and l 2 is ca ptured rather by the notion of spr e ad , which may b e defined as the ratio of tw o quadrances as follows. Suppo se l 1 and l 2 int e r sect at the po in t A. Cho ose a p oint B 6 = A o n o ne of the lines , say l 1 , a nd let C be the fo ot o f the per pendicula r fro m B to l 2 . Then A s B C l l 1 2 Q R Figure 3: Sprea d s b etw een tw o lines l 1 and l 2 the spread s betw een l 1 and l 2 is s = s ( l 1 , l 2 ) = Q ( B , C ) Q ( A, C ) = Q R . 6 This r atio is indep endent of the choice of B , acco rding to Thales, and is defined b etwe en lines, not r ays . P arallel lines are defined to hav e spr ead s = 0 , while per pendicula r lines hav e spread s = 1 . The spread betw ee n the lines l 1 and l 2 with equations a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0 turns out to b e s ( l 1 , l 2 ) = ( a 1 b 2 − a 2 b 1 ) 2 ( a 2 1 + b 2 1 ) ( a 2 2 + b 2 2 ) . (1) Since this is a r ational expressio n, spread b ecomes a useful co ncept also over general fields, although in this paper we stick to the usual situatio n ov er the decimal n umber s. Y ou may chec k that the s pread cor resp onding to 30 ◦ or 150 ◦ or 21 0 ◦ or 33 0 ◦ is s = 1 / 4 , the spread cor resp onding to 45 ◦ or 13 5 ◦ etc. is s = 1 / 2, and the spread corresp onding to 60 ◦ or 120 ◦ etc. is 3 / 4 . The following spr e ad pr otr actor was cr eated b y M. Oss ma nn [1]. Figure 4: A spread protra ctor W e us e the notatio n that a tr iangle A 1 A 2 A 3 has quadrances Q 1 , Q 2 and Q 3 , as well as sprea ds s 1 , s 2 and s 3 , lab elled as in Fig ure 5. Note the diagr ammatic conv entions that help us distinguish these quantities from dis ta nce a nd a ngle. A 1 A 3 A 2 s 1 Q 1 Q 3 Q 2 s 3 s 2 Figure 5: Quadranc e s and spreads of a triangle 7 Rational trigonometry Let’s s e e how to c o m bine the three ancient Greek theorems as re s tated ab ov e to derive the main laws of r ational trig onometry , independent of c lassical tr igonom- etry , and without any need for tra ns cenden ta l functions. If H 3 is the qua drance of the altitude from A 3 to the line A 1 A 2 , then the T riangle ar ea theorem and the definition of spread give area 2 = Q 3 × H 3 4 = Q 3 Q 2 s 1 4 = Q 3 Q 1 s 2 4 . By symmetry , w e g et the following a nalog of the Sine law. Theorem 5 (Spread l a w) F or a triangle with quadr anc es Q 1 , Q 2 and Q 3 , and spr e ads s 1 , s 2 and s 3 , s 1 Q 1 = s 2 Q 2 = s 3 Q 3 = 4 a r ea 2 Q 1 Q 2 Q 3 . By equating the for m ula s for 16 area 2 obtained from the T ria ngle ar ea the- orem and Archim edes’ theorem, w e get 4 Q 2 Q 3 s 1 = ( Q 1 + Q 2 + Q 3 ) 2 − 2  Q 2 1 + Q 2 2 + Q 2 3  = 2 Q 1 Q 2 + 2 Q 1 Q 3 + 2 Q 2 Q 3 − Q 2 1 − Q 2 2 − Q 2 3 . Rearra ng ing gives the following a nalog o f the Co sine law. Theorem 6 (Cross la w) F or a triangle with qu adr anc es Q 1 , Q 2 and Q 3 , and spr e ads s 1 , s 2 and s 3 , ( Q 1 − Q 2 − Q 3 ) 2 = 4 Q 2 Q 3 (1 − s 1 ) . Now substitute Q 1 = s 1 D , Q 2 = s 2 D and Q 3 = s 3 D , wher e D = Q 1 Q 2 Q 3 / 4 a r ea 2 from the Spread law into the Cro ss law, and cancel the common factor of D 2 . The result is the relation ( s 1 − s 2 − s 3 ) 2 = 4 s 2 s 3 (1 − s 1 ) betw een the three spr eads of a tria ngle, which can b e rewritten mo re symmet- rically as follows. Theorem 7 (T riple spread form ula) ( s 1 + s 2 + s 3 ) 2 = 2  s 2 1 + s 2 2 + s 2 3  + 4 s 1 s 2 s 3 . This formula is a defor mation of the T r iple qua d fo rmu la by a single cubic term, and is the a nalog in ratio nal trigo nometry to the classica l fact that the three angles of a triangle sum to 3 . 141 59 2 653 5 9 . . . . The T riple quad formula , Pythagor as’ the or em , the Spr e ad law , the Cr oss law and the T riple spr e ad formula a re the five main laws o f ra tional tr ig onometry . 8 W e now s ee these ar e closely linked to the g e ometrical work of the ancient Greeks. As demonstrated at some length in [2], these for m ula s and a few additional ones suffice to s olve the ma jor it y of trigo nometric pro blems, usua lly more simply , more accurately and more elegan tly than the classical theory in volving transcen- dent a l circular functions and their in verses. As shown in [3] and [5], the same formulas ex tend to geo metry o ver arbitrary fields (not of c har acteristic t wo) and with general quadratic forms. In retrosp ect, the blind sp ot firs t o ccur red with the Pythago reans, who ini- tially b elieved that all of natur e should b e expressible in terms o f natural num- ber s and their prop ortio ns . When they discovered tha t the ratio of the length o f a diag onal to the length of a side of a squar e was the incommensurable pr op or- tion √ 2 : 1 , legend has it that they tossed the exp oser of the secret ov er bo a rd while at sea. Had they maint a ined their b eliefs in the workings of the Divine Mind, and stuck with the squar es of the lengths as the crucial quantities in ge ometry, then mathematics would have had a significantly different history , Einstein’s sp ecial theory of relativity w ould p oss ibly hav e b een disc ov ered earlier, algebraic geom- etry would hav e quite another as p ect, a nd students w ould today be s tudying a simpler and more elegant trigono metry—muc h mor e happily! References [1] M. Ossmann, ‘Print a Protractor’, do wnloa d online at ht tp:/ / www.ossmann.com/ protractor / . [2] N. J. Wildb erger , Divine Pr op ortions: R ational T rigonometry to U niversal Ge ometry , Wild Egg Bo oks, Sydney , 2005, ht tp:/ / wildegg.com. [3] N. J. Wildberger, Affine and Pro jective Ra tional T rigo no metry , 2006 arXiv:math/06 12499 . [4] N. J. Wildb erge r, A Ratio nal Approach to T rig onometry , Math H orizons , Nov. 2007 . [5] N. J. Wildb erger, One dimensional metrical geometry , Ge ometriae De dic ata , 128 , no. 1, 145-166 , 2007. 9