To express the reciprocals of the perpendiculars let fall from the angles on the opposite sides in terms of the radii of the inscribed and exscribed circles.

p", p",

Let the perpendiculars on the sides a, b, c, be denoted by p', p", add the last two equations of Proposition H, and we obtain

If we subtract each of the equations in Prop. H from the equation

PROPOSITION J.

To express the reciprocals of the radii of the inscribed and exscribed circles in terms of the perpendiculars.

Add together the three last equations, and we obtain the relation

PROPOSITION K.

To express the area of a triangle in terms of the perpendiculars.

By multiplying together the equations of the last Proposition, we get

To express the radius of the circumscribed circle in terms of the sides.

By exercise 147 (Books I. II. III.),

BOOK FIFTH.

DEFINITION I.

Ὅρος α'.—Μέρος ἐστὶ μέγεθος μεγέθους, τὸ ἔλασσον τοῦ μείζονος, ὅταν καταμετρῇ τὸ μεῖζον.

A part is a magnitude of a magnitude, the less of the greater, when the less measures the greater.

Annotation.

What is here called a part is called by the moderns a submultiple. When the less is contained in the greater an exact number of times without remainder, it is said to measure the greater, and to be a submultiple of it.

Thus, 3 and 7 are submultiples of 21; x, y, z are submultiples of

DEFINITION II.

Ὅρος β'.—Πολλαπλάσιον δὲ τὸ μεῖζον τοῦ ἐλάσσονος, ὅταν καταμετρῆται ὑπὸ τοῦ ἐλάττονος.

The greater magnitude is a multiple of the less, when it is measured by the less.

Annotation.

Thus, 21 is a multiple of 7, and so is 63; xy is a multiple of x, and also of y.

DEFINITION III.

Ὅρος γ ́.—Λόγος ἐστὶ δύο μεγεθῶν ὁμογενῶν ἡ κατὰ πηλικότητα πρὸς ἄλληλα ποιὰ σχέσις.

Ratio is the mutual relation (habitude) of two magnitudes of the same kind, each to other, with respect to quantity.

Annotation.

In every ratio, that quantity which is referred to another quantity is called the antecedent of the ratio, and that to which the other is referred is called the consequent of the ratio, as in the ratio 6: 4; 6 is the antecedent, and 4 the consequent.

The quantity of any ratio is known by dividing the antecedent by

the consequent; as the ratio of 12: 5 is expressed by

12

5

; or the

i. e. the ratio of a to b is less than, equal to, or greater than the ratio of c to d.-Isaac Barrow.

DEFINITION IV.

"Ορος δ'.--Αναλογία δὲ, ἡ τῶν λόγων ταυτότης.

Proportion (analogy) is the identity (sameness) of ratios.

Annotation.

If the ratio of a: b be the same as that of c: d, then the four terms taken together form a proportion, which is thus expressed— abc: d.

DEFINITION V.

Ὅρος ε'.—Λόγον ἔχειν πρὸς ἄλληλα μεγέθη λέγεται, ἃ δύναται πολλαπλασιαζόμενα ἀλλήλων ὑπερέχειν.

Magnitudes are said to have a ratio to each other, which, being multiplied, may exceed, each the other.

Annotation.

The object of this definition is to exclude infinitely great and infinitely small quantities from comparison with finite quantities. Two infinitely great quantities may have a relation to each other, and so may two infinitely small quantities; but neither can have a ratio to any finite quantity.

DEFINITION VI.

Ορος 5'.—Εν τῷ ἀυτῷ λόγῳ μεγέθη λέγεται εἶναι, πρῶτον πρὸς δέυτερον καὶ τρίτον πρὸς τέταρτον, ὅταν τὰ τοῦ πρώτου καὶ τρίτου ἰσάκις πολλαπλάσια, τῶν τοῦ δευτέρου καὶ τετάρτου ἰσάκις πολλαπλασίων, καθ ̓ ὁποιονοῦν πολλαπλασιασμον, ἑκατέρον ἑκατέρου ἢ ἅμα ὑπερέχῃ, ἢ ἅμα ἴσα ᾖ, ἢ ἅμα ἐλλείπῃ ληφθέντα κατάλληλα.

Magnitudes are said to be in the same ratio, the first to the second as the third to the fourth; when any equimultiples whatsoever of the first and third, compared with any equimultiples whatsoever of the second and fourth, each with each, are greater together, or equal together, or less together.

Annotation.

Two magnitudes are said to be commensurable, when they have a common multiple; i. e. when some multiple of the first is equal to some multiple of the second.

Thus x and y are commensurable magnitudes, when two whole numbers m and n can be found such that

mx = ny.

If x and y be whole numbers themselves, this is always possible; and therefore all whole numbers are commensurable.

But there are many magnitudes which have no common multiple, and are therefore said to be incommensurable; for example

1. The side of a square, and its diagonal.

2. The diameter of a circle, and its periphery.

3. The side of a cube, and its diagonal.

4. The segments of a line cut as in II. II.

As all these quantities, although incommensurable, yet have a definite ratio to each other, a definition of equality of ratios has been here devised by Euclid, which includes all magnitudes, and which constitutes a test easy of application, and superior to any other which his commentators have proposed to substitute for it.

The following illustration will show the meaning of the definition : Let

x: y = a:b,

then, if any whole numbers whatsoever m and n be assumed; taking mx and ma; ny and nb;