peculiarly facilitate the application of algebra to geometrical investigations.

(1.) "The sides about the equal angles of equiangular triangles are proportionals," &c.

(2.) "If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means," &c.

52. Specify those propositions of Book vi. which generalize certain propositions in Book i.

Prop. iv. v. vi. Book vi., generalize Prop. xxvi. viii. iv. Book i.

53. Of what does the sixth Book of Euclid treat?

Of the sides and areas of certain rectilineal figures, and contains the investigation of lines that have a proposed ratio to given lines.

54. What do the eleventh and twelfth Books treat of? Of the geometry of planes and solids.

55. Of how many Books do the Elements of Euclid consist?

Fifteen.

56. About what period did Euclid flourish?

About 300 years, B. C. at the time Ptolemy Lagos was king of Egypt.

57. What answer did Euclid give to the question of his pupil, King Ptolemy-" Is there no shorter way of coming at geometry than by your Elements ?”

"There is no ROYAL road to GEOMETRY."

346

NOTES.

12. 1.

* 7.2.

NOTE I.

ANOTHER DEMONSTRATION OF PROPOSITION XIII.

BOOK II.

(See the figures in p. 64.)

Let ABC be any triangle, and the angle at B one of its acute angles; and upon BC, or its prolongation, one of the sides containing it, let fall the perpendicular* AD from the opposite angle: the square of the side AC, subtending the angle B, shall be less than the squares of AB, BC, by twice the rectangle CB, BD.

For if the perpendicular AD falls within the triangle ABC, the straight line BC is divided into two parts in the point D, and if it falls without the triangle, then BD is divided into two parts in the point C; therefore, in either case, the squares of CB, BD are equal to twice the rectangle contained by CB, BD, together with the square of DC: to each of these equals add the square of AD; therefore, the squares of CB, BD, DA, * 2. Ax. are equal to twice the rectangle CB, BD, together with the squares of AD, DC; but the square of AB is * 47. 1. equal to the squares of BD, DA, because the angle BDA is a right angle; and the square of AC is equal to the squares of AD, DC; therefore, the squares of CB, BA are equal to the square AC, and twice the rectangle CB, BD; that is, the square of AC alone, is less than the squares of CB, BA, by twice the rectangle CB, BD. But if the side AC be perpendicular to BC, then BC is the straight line between the perpendicular

and the acute angle at B: and because the square of AB is equal to the squares of BC, CA; therefore the 47. 1.

squares of AB, BC are equal to the square of AC, and twice the square of BC: therefore in every triangle &c. Q. E. D.

NOTE II.

NUMERICAL EXPLANATion of defiNITION V. BOOK V.

1. Let it be required to ascertain whether 4 has the same ratio to 7 which 24 has to 42, or if 4 : 7 :: 24: 42.

Of the first and third take any equimultiples 12 and 72; and of the second and fourth take any equimultiples 14 and 84; then the multiples of the four proposed numbers, taken in their order, are

12, 14, 72, 84.

Now the multiple of the first is less than that of the second, and the multiple of the third is less than that of the fourth.

Again, of the first and third take any equimultiples 28 and 168, and of the second and fourth take the equimultiples 28 and 168; then the multiples are

28, 28, 168, 168;

where the multiple of the first is equal to that of the second, and the multiple of the third is also equal to that of the fourth.

Lastly, of the first and third take any equimultiples 20 and 120, and of the second and fourth take equimultiples 14 and 84; then these multiples are

20, 14, 120, 84.

Here the multiple of the first is greater than that of the second, and the multiple of the third is also greater

than that of the fourth; therefore the three conditions enunciated in the definition are completely fulfilled and consequently

4:7: 24: 42;

or 4 has to 7 the same ratio which 24 has to 42.

2. Determine, by Euclid's definition of proportion, whether

7:22: 113: 355.

The failure of fulfilling any one of the three conditions stated in the definition, will be a certain indication that the four given numbers cannot constitute a proportion. Now, if by taking equimultiples of the first and third we have

154, 22, 2486, 355,

and by taking equimultiples of the second and fourth we have

154, 154, 2486, 2485;

then the multiple of the first is equal to that of the second, but the multiple of the third is not equal to that of the fourth; consequently the proposed numbers cannot constitute a proportion, because they fail to fulfil one of the conditions of the definition of proportion.