BOOK II.

DEFINITIONS.

1. Any rectangular parallelogram is said to be contained by the two straight lines containing the right angle.

2. And in any parallelogrammic area let any one whatever of the parallelograms about its diameter with the two complements be called a gnomon.

DEFINITION I.

Πᾶν παραλληλόγραμμον ὀρθογώνιον περιέχεσθαι λέγεται ὑπὸ δύο τῶν τὴν ὀρθὴν γωνίαν περιεχουσῶν εὐθειῶν.

As the full expression in Greek for "the angle BAC" is "the angle contained by the (straight lines) ΒΑ, AC,” ή ὑπὸ τῶν ΒΑ, ΑΓ περιεχομένη yovía, so the full expression for "the rectangle contained by BA, AC" ἰς τὸ ὑπὸ τῶν ΒΑ, ΑΓ περιεχόμενον ὀρθογώνιον. In both cases the substantive and participle can be omitted because the feminine or neuter of the article enables us to distinguish whether an angle or a rectangle is meant; but the difference in Euclid's phraseology is that the words imò Tv BA, Ar appear always in full for the rectangle, whereas the shorter vò BAT is used in describing the angle. Archimedes and Apollonius, on the other hand, frequently use the expression Tò mò BAT for the rectangle BA, AC, just as they use ò BAT for the angle BAC.

DEFINITION 2.

Παντὸς δὲ παραλληλογράμμου χωρίου τῶν περὶ τὴν διάμετρον αὐτοῦ παραλλη λογράμμων ἐν ὁποιονοῦν σὺν τοῖς δυσὶ παραπληρώμασι γνώμων καλείσθω.

Meaning literally a thing enabling something to be known, observed or verified, a teller or marker, as we might say, the word gnomon (ywμwr) was first used in the sense (1) in which it appears in a passage of Herodotus (11. 109) stating that "the Greeks learnt the rodos, the gnomon and the twelve parts of the day from the Babylonians." According to Suidas, it was Anaximander (611-545 B.C.) who introduced the gnomon into Greece. Whatever may be the details of the construction of the two instruments called the wóλos and the gnomon, so much is certain, that the gnomon had to do with the

66

measurement of time by shadows thrown by the sun, and that the word signified the placing of a staff perpendicular to the horizon. This is borne out by the statement of Proclus that Oenopides of Chios, who first investigated the problem (Eucl. 1. 12) of drawing a perpendicular from an external point to a given straight line, called the perpendicular a straight line drawn "gnomon-wise" (Kaтà yváμova). Then (2) we find the term used of a mechanical instrument for drawing right angles, as shown in the figure annexed. This seems to be the meaning in Theognis 805, where it is said that the envoy sent to consult the oracle at Delphi should be “ straighter (ἰθύτερος) than the τόρνος, the στάθμη and the gnomon," and all three words evidently denote appliances, the Topvos being an instrument for drawing a circle (probably a string stretched between a fixed and a moving point), and the orálun a plumb-line. Next (3) it was natural that the gnomon, owing to its shape, should become the figure which remained of a square when a smaller square was cut out of one corner (or the figure, as Aristotle says, which when added to a square increases its size but does not alter its form). We have seen (note on 1. 47, p. 351) that the Pythagoreans used the term in this sense, and further applied it, by analogy, to the series of odd numbers as having the same property in relation to square numbers. The earliest evidence for this is the fragment of Philolaus (c. 460 B.C.) already mentioned (see Boeckh, Philolaos des Pythagoreers Lehren, p. 141) where he says that "number makes all things knowable and mutually agreeing (ποτάγορα ἀλλάλοις) in the way characteristic of the gnomon" (kaтà yvwμovos púσi). As Boeckh says (p. 144), it would appear from the fragment that the connexion between the gnomon and the square to which it is added was regarded as symbolical of union and agreement, and that Philolaus used the idea to explain the knowledge of things, making the knowing embrace and grasp the known as the gnomon does the square. Scholium II. No. 11 (Euclid, ed. Heiberg, Vol. v. p. 225), which says "It is to be noted that the gnomon was discovered by geometers with a view to brevity, while the name came from its incidental property, namely that from it the whole is known, whether of the whole area or of the remainder, when it is either placed round or taken away. In sundials too its sole function is to make the actual time of day known."

Cf.

The geometrical meaning of the word is extended in the definition of gnomon given by Euclid, where (4) the gnomon has the same relation to any parallelogram as it before had to a square. From the fact that Euclid says "let" the figure described "be called a gnomon" we may infer that he was using the word in the wider sense for the first time. Later still (5) we find Heron of Alexandria (1st cent. A.D.) defining a gnomon in general as any figure which, when added to any

figure whatever, makes the whole figure similar to that to which it is added. In this definition of Heron (Def. 59) Hultsch brackets the words which make it apply to any number as well; but Theon of Smyrna, who explains that plane, triangular, square, solid and other kinds of numbers are so called after the likeness of the areas which they measure, does make the term in its most general sense apply to numbers. "All the successive numbers which [by being successively added] produce triangles or squares or polygons are called gnomons" (p. 37, 11-13, ed. Hiller). Thus the successive odd numbers added

together make square numbers; the gnomons in the case of triangular numbers are the successive numbers 1, 2, 3, 4...; those for pentagonal numbers are the series 1, 4, 7, 10... (the common difference being 3), and so on. In general, the successive gnomonic numbers for any polygonal number, say of n sides, have n 2 for their common difference (Theon of Smyrna, P. 34, 13-15).

GEOMETRICAL ALGEBRA.

We have already seen (cf. part of the note on I. 47 and the above note on the gnomon) how the Pythagoreans and later Greek mathematicians exhibited different kinds of numbers as forming different geometrical figures. Thus, says Theon of Smyrna (p. 36, 6-11), "plane numbers, triangular, square and solid numbers, and the rest, are not so called independently (vpiws) but in virtue of their similarity to the areas which they measure; for 4, since it measures a square area, is called square by adaptation from it, and 6 is called oblong for the same reason." A "plane number" is similarly described as a number obtained by multiplying two numbers together, which two numbers are sometimes spoken of as "sides," sometimes as the "length" and "breadth" respectively, of the number which is their product.

The product of two numbers was thus represented geometrically by the rectangle contained by the straight lines representing the two numbers respectively. It only needed the discovery of incommensurable or irrational straight lines in order to represent geometrically by a rectangle the product of any two quantities whatever, rational or irrational; and it was possible to advance from a geometrical arithmetic to a geometrical algebra, which indeed by Euclid's time (and probably long before) had reached such a stage of development that it could solve the same problems as our algebra so far as they do not involve the manipulation of expressions of a degree higher than the second. In order to make the geometrical algebra so generally effective, the theory of proportions was essential. Thus, suppose that x, y, z etc. are quantities which can be represented by straight lines, while a, B, y etc. are coefficients which can be expressed by ratios between straight lines. We can then by means of Book vi. find a single straight line d such that

where a represents any ratio between straight lines, also requires recourse to the sixth Book, though, e.g., if a isor or any submultiple of unity, or if a is 2, 4 or any power of 2, we should not require anything beyond Book 1. for solving the equation. Similarly the general form of a quadratic equation requires Book VI. for its geometrical solution, though particular quadratic equations may be so solved by means of Book 11. alone.

Besides enabling us to solve geometrically these particular quadratic equations, Book II. gives the geometrical proofs of a number of algebraical formulae. Thus the first ten propositions give the equivalent of the several identities

The form of these identities may of course be varied according to the different symbols which we may use to denote particular portions of the lines given in Euclid's figures. They are, for the most part, simple identities, but there is no reason to suppose that these were the only applications of the geometrical algebra that Euclid and his predecessors had been able to make. We may infer the very contrary from the fact that Apollonius in his Conics frequently states without proof much more complicated propositions of the kind.

It is important however to bear in mind that the whole procedure of Book 11. is geometrical; rectangles and squares are shown in the figures, and the equality of certain combinations to other combinations is proved by those figures. We gather that this was the classical or standard method of proving such propositions, and that the algebraical method of proving them, with no figure except a line with points marked thereon, was a later introduction. Accordingly Eutocius' method of proving certain lemmas assumed by Apollonius (Conics, 11. 23 and 11. 29) probably represents more nearly than Pappus' proof of the same the point of view from which Apollonius regarded them.

It would appear that Heron was the first to adopt the algebraical method of demonstrating the propositions of Book II., beginning from the second, without figures, as consequences of the first proposition corresponding to

a (b + c + d) = ab + ac + ad.

According to an-Nairīzī (ed. Curtze, p. 89), Heron explains that it is not possible to prove II. I without drawing a number of lines (i.e. without actually drawing the rectangles), but that the following propositions up to II. 10 inclusive can be proved by merely drawing one line. He distinguishes two varieties of the method, one by dissolutio, the other by compositio, by which he seems to mean splitting-up of rectangles and squares, and combination of them into others. But in his proofs he sometimes combines the two varieties.

When he comes to 11. 11, he says that it is not possible to do without a figure because the proposition is a problem, which accordingly requires an operation and therefore the drawing of a figure.

The algebraical method has been preferred to Euclid's by some English editors; but it should not find favour with those who wish to preserve the

essential features of Greek geometry as presented by its greatest exponents, or to appreciate their point of view.

It may not be out of place to add a word with reference to the geometrical equivalent of the algebraical operations. The addition and subtraction of quantities represented in the geometrical algebra by lines is of course effected by producing the line to the required extent or cutting off a portion of it. The equivalent of multiplication is the construction of the rectangle of which the given lines are adjacent sides. The equivalent of the division of one quantity represented by a line by another quantity represented by a line is simply the statement of a ratio between lines on the principles of Books v. and vi. The division of a product of two quantities by a third is represented in the geometrical algebra by the finding of a rectangle with one side of a given length and equal to a given rectangle or square. This is the problem of application of areas solved in 1. 44, 45. The addition and subtraction of products is, in the geometrical algebra, the addition and subtraction of rectangles or squares; the sum or difference can be transformed into a single rectangle by means of the application of areas to any line of given length, corresponding to the algebraical process of finding a common measure. Lastly, the extraction of the square root is, in the geometrical algebra, the finding of a square equal to a given rectangle, which is done in II. 14 with the help of 1. 47.