Indian Mathematics/Circ. A.D. 600–1200

by George Rusby Kaye

IV.

A.D. 600—1200.

11. Āryabhata appears to have given a definite bias to Indian mathematics, for following him we have a series of works dealing with the same topics. Of the writers themselves we know very little indeed beyond the mere names but some if not all the works of the following authors have been preserved:

Brahmagupta‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥born A.D. 598.

Mahāvīra‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥? 9th century.

S'rīdhara‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥born A.D. 991.

Bhāskara‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥born A.D. 1114.

Bhāskara is the most renowned of this school, probably undeservedly so, for Brahmagupta's work is possibly sounder mathematically and is of much more importance historically. Generally these writers treat of the same topics—with a difference—and Brahmagupta's work appears to have been used by all the others. Bhāskara mentions another mathematician, Padmanābha, but omits from his list Mahāvīra.

One of the chief points of difference is in the treatment of geometry. Brahmagupta deals fairly completely with cyclic quadrilaterals while the later writers gradually drop this subject until by the time of Bhāskara it has ceased to be understood.

The most interesting characteristics of the works of this period are the treatment of:

(i) indeterminate equations;

[(ii) the rational right-angled triangle;

and (iii) the perfunctory treatment of pure geometry.

Of these topics it will be noted that the second was dealt with to some extent in the S'ulvasūtras; but a close examination seems to show that there is no real connection and that the writers of the third period were actually ignorant of the results achieved by Baudhāyana and Āpastamba.

12. Indeterminate Equations. The interesting names and dates connected with the early history of indeterminates in India are:

cir. A.D.

Diophantus‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥360

Hypatia‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥400

Āryabhata‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥born  476

Brahmagupta‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ "⁠ 598

Bhāskara‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ "⁠1114

That we cannot fill up the gap between Diophantus and Āryabhata with more than the mere name of Hypatia is probably due to the fanatic ignorance and cruelty of the early Alexandrian Christians rather than the supposed destruction of the Alexandrian library by the Muhammadans. It would be pleasant to conceive that in the Indian works we have some record of the advances made by Hypatia, or of the contents of the lost books of Diophantus—but we are not justified in indulging in more than the mere fancy. The period is one of particular interest. The murder of Hypatia (A.D. 415), the imprisonment and execution of Boethius (A.D. 524), the closing of the Athenean schools in A.D. 530 and the fall of Alexandria in 640 are events full of suggestions to the historian of mathematics. It was during this period also that Damascius, Simplicius (mathematicians of some repute) and others of the schools of Athens, having heard that Plato's ideal form of government was actually realised under Chosroes I in Persia, emigrated thither (c. A.D. 532). They were naturally disappointed but the effect of their visit may have been far greater than historical records show.

13. The state of knowledge regarding indeterminate equations in the west at this period is not definitely known. Some of the works of Diophantus and all those of Hypatia are lost to us; but the extant records show that the Greeks had explored the field of this analysis so far as to achieve rational solutions (not necessarily integral) of equations of the first and second degree and certain cases of the third degree. The Indian works record distinct advances on what is left of the Greek analysis. For example they give rational integral solutions of ${\begin{aligned}\\&\scriptstyle {(A)\quad ax\pm by=c}\\&\scriptstyle {(B)\quad Du^{2}+1=t^{2}}\\\end{aligned}}$

{{p|aj}The solution of (A) is only roughly indicated by Āryabhata but Brahmagupta's solution (for the positive sign) is practically the same as $\scriptstyle {x=\pm cq-bt,\quad y=\mp cp+at}$ where t is zero or any integer and p/q is the penultimate convergent of a/b.

The Indian methods for the solution of $\scriptstyle {Du^{2}+1=t^{2}}$ may be summarised as follows:

If $\scriptstyle {Da^{2}+b=c^{2}}$ and $\scriptstyle {D\alpha ^{2}+\beta =\gamma ^{2}}$ then will ${\begin{aligned}\\&\scriptstyle {(a)\quad D(c\alpha \mp \gamma a)^{2}+b\beta =(c\gamma \pm a\alpha D)^{2}}\\&\scriptstyle {(b)\quad D({\frac {c+\alpha r}{b}})^{2}\pm {\frac {r^{2}-D}{b}}=({\frac {Da+cr}{b}})^{2}}\\\end{aligned}}$ where r is any suitable integer.

Also $\scriptstyle {(c)\quad D({\frac {2n}{D-n^{2}}})^{2}+1=({\frac {D+n^{2}}{D-n^{2}}})^{2}}$ where n is any assumed number.

The complete integral solution is given by a combination (a) and (b) of which the former only is given by Brahmagupta, while both are given by Bhāskara (five centuries later). The latter designates (a) the 'method by composition' and (b) the 'cyclic method.' These solutions are alone sufficient to give to the Indian works an important place in the history of mathematics. Of the 'cyclic method' (i.e., the combination of (a) and (b)) Hankel says, "It is beyond all praise: it is certainly the finest thing achieved in the theory of numbers before Lagrange." He attributed its invention to the Indian mathematicians, but the opinions of the best modern authorities (e.g., Tannery, Cantor, Heath) are rather in favour of the hypothesis of ultimate Greek origin.

The following conspectus of the indeterminate problems dealt with by the Indians will give some idea of their work in this direction; and although few of the cases actually occur in Greek works now known to us the conspectus significantly illustrates a general similarity of treatment.

^[1](1) $\scriptstyle {ax\pm by=c}$
(2) $\scriptstyle {ax+by+cz+\cdots =l}$
(3) $\scriptstyle {x\equiv a_{1}Mod.b_{1}\equiv \dots \equiv a_{a}Mod.b_{4}}$
(4) $\scriptstyle {Ax+By+Cxy=D}$
(5) $\scriptstyle {Du^{2}+1=t^{2}}$
(6) $\scriptstyle {Du^{2}-1=t^{2}}$
(7) $\scriptstyle {Du^{2}\pm s=t^{2}}$
(8) $\scriptstyle {D^{2}u^{2}\pm s=t^{2}}$
(9) $\scriptstyle {u^{2}+s=at^{2}}$
(10) $\scriptstyle {Du^{2}\pm au=t^{2}}$
(11) $\scriptstyle {s-Du^{2}=t^{2}}$
(12) $\scriptstyle {Du+s=t^{2}}$
(13) $\scriptstyle {x\pm a=s^{2},\quad x\pm b=t^{2}}$
(14)$ax+1=s^{2},\quad bx+a=t^{2}$
(15)$2(x^{2}-y^{2})+3=s^{2},\quad 3(x^{2}-y^{2})+3=t^{2}$
(16)$ax^{2}+by^{2}=s^{2},\quad ax^{2}-by^{2}+1=t^{2}$
(17)$x^{2}+y^{2}\pm 1=s^{2},\quad x^{2}-y^{2}\pm 1=t^{2}$
(18)$x^{2}-a\equiv x^{2}-b\equiv o~Mod.~c$
(19)$ax^{2}+b\equiv o~Mod.~c$
(20)$x+y=s^{2},\quad x-y=t^{2},\quad xy=u^{3}$
(21)$x^{3}+y^{3}=s^{2},\quad x^{2}+y^{2}=t^{3}$
(22)$x-y=s^{3},\quad x^{2}+y^{2}=t^{3}$
(23)$x+y=s^{2},\quad x^{3}+y^{2}=t^{2}$
(24)$\scriptstyle {x^{3}+y^{2}+xy=s^{2},\quad (x+y)s+1=t^{2}}$
(25)$\scriptstyle {ax+1=s^{3},\quad as^{2}+1=t^{2}}$
(26)$\scriptstyle {wxyz=a(w+x+y+z)}$
(27)$\scriptstyle {x^{3}-a\equiv o~Mod.~b}$
(28)${\begin{aligned}\\&\scriptstyle {x+y+3=s^{2},~x-y+3=t^{2},~x^{2}+y^{2}-4=u^{2},}\\&\scriptstyle {x^{2}-y^{2}+12=v^{2},\ {\frac {xy}{2}}+y=w^{3},}\\&\scriptstyle {s+t+u+v+w+2=z^{2}}\\\end{aligned}}$
(29)$\scriptstyle {{\frac {\sqrt[{3}]{xy+y}}{2}}+{\sqrt {x^{2}+y^{2}}}+{\sqrt {x+y+2}}+{\sqrt {x-y+2}}+{\sqrt {x^{2}-y^{2}+8}}=t^{2}}$
(30)${\begin{aligned}\\&\scriptstyle {w+2=a^{2},~x+2=b^{2},~y+2=c^{2},~z+2=d^{2},}\\&\scriptstyle {wx+18=e^{2},\ xy+18=f^{2},\ yz+18=g^{2},}\\&\scriptstyle {a+b+c+d+e+f+g+11=13}\\\end{aligned}}$.

14. Rational right-angled triangles.—The Indian mathematicians of this period seem to have been particularly attracted by the problem of the rational right-angled triangle and give a number of rules for obtaining integral solutions. The following summary of the various rules relating to this problem shows the position of the Indians fairly well.

	A	B	$\scriptstyle {\sqrt {A^{2}+B^{2}}}$	Authorities.
i	n	$\scriptstyle {\frac {n^{2}-1}{2}}$	$\scriptstyle {\frac {n^{2}+1}{2}}$	Pythagoras (according to Proclus)
ii	$\scriptstyle {\sqrt {mn}}$	$\scriptstyle {\frac {mn}{2}}$	$\scriptstyle {\frac {m+n}{2}}$	Plato (according to Proclus)
iii	$\scriptstyle {2mn}$	$\scriptstyle {m^{2}-n^{2}}$	$\scriptstyle {m^{2}+n^{2}}$	Euclid and Diophantus.
iv	$\scriptstyle {\frac {m(n^{2}-1)}{n^{2}+1}}$	$\scriptstyle {\frac {2mn}{n^{2}+1}}$	m	Diophantus.
v	2mn	$\scriptstyle {m^{2}-n^{2}}$	$\scriptstyle {m^{2}+n^{2}}$	Brahmagupta and Mahāvīra.
vi	$\scriptstyle {\sqrt {m}}$	$\scriptstyle {{\frac {1}{2}}({\frac {m}{n}}-n)}$	$\scriptstyle {{\frac {1}{2}}({\frac {m}{n}}+n)}$	Brahmagupta, Mahāvīra, and Bhāskara.
vii	m	$\scriptstyle {\frac {2mn}{n^{2}-1}}$	$\scriptstyle {\frac {m(n^{2}+1)}{n^{2}-1}}$	Bhāskara.
viii	$\scriptstyle {\frac {m(m^{2}-1)}{n^{2}+1}}$	$\scriptstyle {\frac {2mn}{n^{2}+1}}$	m	Bhāskara.
ix	2lmn	$\scriptstyle {l(m^{2}-n^{2})}$	$\scriptstyle {l(m^{2}+n^{2})}$	General formula.

Mahāvīra gives many examples in which he employs formula (V) of which he terms m and n the 'elements.' From given elements he constructs triangles and from given triangles he finds the elements, e.g., "What are the 'elements' of the right-angled triangle (48, 55, 73)? Answer: 3, 8."

Other problems connected with the rational right-angled triangle given by Bhāskara are of some historical interest: e.g., (1) The sum of the sides is 40 and the area 60, (2) The sum of the sides is 56 and their product $\scriptstyle {7\times 600}$, (3) The area is numerically equal to the hypotenuse, (4) The area is numerically equal to the product of the sides.

15. The geometry of this period is characterised by:

(1) Lack of definitions, etc.;

(2) Angles are not dealt with at all;
(3) There is no mention of parallels and no theory of proportion;
(4) Traditional inaccuracies are not uncommon;

(5) A gradual decline in geometrical knowledge is noticeable.

On the other hand, we have the following noteworthy rules relating to cyclic quadrilaterals—${\begin{aligned}\\\scriptstyle {(i)\ }&\scriptstyle {Q={\sqrt {(s-a)(s-b)(s-c)(s-d)}}}\\\scriptstyle {(ii)\ }&\scriptstyle {x^{2}=(ad+bc)(ac+bd)(ab+cd)}\\&\scriptstyle {y^{2}=(ab+cd)(ac+bd)(ad+bc)}\\\end{aligned}}$where x and y are the diagonals of the cyclic quadrilateral (a, b, c, d). This (ii) is sometimes designated as 'Brahmagupta's theorem'.

16. The absence of definitions and indifference to logical order sufficiently differentiate the Indian geometry from that of the early Greeks; but the absence of what may be termed a theory of geometry hardly accounts for the complete absence of any reference to parallels and angles. Whereas on the one hand the Indians have been credited with the invention of the sine function, on the other there is no evidence to show that they were acquainted with even the most elementary theorems (as such) relating to angles.

The presence of a number of incorrect rules side by side with correct ones is significant. The one relating to the area of triangles and quadrilaterals, viz., the area is equal to the product of half the sums of pairs of opposite sides, strangely enough occurs in a Chinese work of the 6th century as well as in the works of Ahmes, Brahmagupta, Mahāvīra, Boethius and Bede. By Mahāvīra, the idea on which it is based—that the area is a function of the perimeter—is further emphasized. Āryabhata gives an incorrect rule for the volume of a pyramid; incorrect rules for the volume of a sphere are common to Āryabhata, S'rīdhara and Mahāvīra. For cones all the rules assume that $\scriptstyle {\pi =3}$. Mahāvīra gives incorrect rules for the circumference and area of an ellipse and so on.

17. Brahmagupta gives a fairly complete set of rules dealing with the cyclic quadrilateral and either he or the mathematician from whom he obtained his material had a definite end in view—the construction of a cyclic quadrilateral with rational elements.—The commentators did not fully appreciate the theorems, some of which are given in the works of Mahāvīra and S'rīdhara; and by the time of Bhāskara they had ceased to be understood. Bhāskara indeed condemns them outright as unsound. "How can a person" he says "neither specifying one of the perpendiculars, nor either of the diagonals, ask the rest? Such a questioner is a blundering devil and still more so is he who answers the question."

Besides the two rules (i) and (ii) already given in paragraph 15, Brahmagupta gives rules corresponding to the formula

(iii) $\scriptstyle {2r={\frac {A}{\sin A}}}$, etc., and

(iv) If $\scriptstyle {a^{2}+b^{2}=c^{2}}$ and $\scriptstyle {\alpha ^{2}+\beta ^{2}=\gamma ^{2}}$ then the quadrilateral $\scriptstyle {(a\gamma ,~c\beta ,~b\gamma ,~c\alpha )}$ is cyclic and has its diagonals at right angles.

This figure is sometimes termed "Brahmagupta's trapezium." From the triangles (3, 4, 5) and (5, 12, 13) a commentator obtains the quadrilateral (39, 60, 52, 25), with diagonals 63 and 56, etc. He also introduces a proof of Ptolemy's theorem and in doing this follows Diophantus (iii, 19) in constructing from triangles (a, b, c) and $\scriptstyle {(\alpha ,~\beta ,~\gamma ,)}$ new triangles $\scriptstyle {(a\gamma ,~b\gamma ,~c\gamma ,)}$ and $\scriptstyle {(\alpha c,~\beta c,~\gamma c,)}$ and uses the actual examples given by Diophantus, namely (39, 52, 65) and (25, 60, 65).

18. An examination of the Greek mathematics of the period immediately anterior to the Indian period with which we are now dealing shows that geometrical knowledge was in a state of decay. After Pappus (c. A.D. 300) no geometrical work of much value was done. His successors were, apparently, not interested in the great achievements of the earlier Greeks and it is certain that they were often not even acquainted with many of their works. The high standard of the earlier treatises had ceased to attract, errors crept in, the style of exposition deteriorated and practical purposes predominated. The geometrical work of Brahmagupta is almost what one might expect to find in the period of decay in Alexandria. It contains one or two gems but it is not a scientific exposition of the subject and the material is obviously taken from western works.

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1. Of these only numbers 1—5, 7, 8, 12—14 occur before the twelfth century.