tionals according to def. 7.; I. 27. being cited instead of I. 25. Therefore, &c.

Cor. 1. In like manner it may be shown that triangles which have the same or equal bases are to one another as their altitudes.

Cor. 2. Also any two triangles are to one another in the ratio which is compounded of the ratios of their bases and of their altitudes.

Cor. 3. Wherefore if the base of one triangle be to the base of another as the altitude of that other to the altitude of the first, the two triangles will be equal to one another (11. Cor. 2.).

PROP. 40.

Triangles which have one angle of the one equal to one angle of the other, are to one another in the ratio which is compounded of the ratios of the sides about the equal angles.

For, if the triangles be completed into parallelograms having the same equal angles and the same sides containing them, these parallelograms (36. Cor. 1.) will be to one another in the ratio which is compounded of the ratios of the sides; therefore the triangles, which are their halves, will be to one another in the same ratio. (17. Cor. 1.)

Otherwise: As in 36. it is demonstrated of the rectangles, and in 36. Cor. 1. of the parallelograms, so here it may, after the same manner, be demonstrated of the two triangles A B C, EBF, by making the equal angles coincide, as at B, and completing the triangle EB C, that the triangle A B C is to the triangle E B F in the ratio which is compounded of the ratios of A B to E B and of B C to BF; 39. being cited instead of 35. Therefore, &c.

E

Cor. Triangles, which have one angle of the one equal to one angle of the other, are to one another as the rectangles under the sides about the equal angles

(37.).

PROP. 41. (Euc. vi. 15.) Triangles which have one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another; and, conversely, equal triangles which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional,

For, if the triangles be completed into parallelograms having the same equal angles and the same sides containing them, these parallelograms (38. Cor. 3.) will be equal to one another, because they have the sides about the equal angles reciprocally proportional; and therefore the triangles, which are their halves, are likewise equal (I. ax. 5.).

And in like manner the converse from the converse part of the same 38. Cor. 3.

Otherwise: As of the rectangles in Prop. 38., so here it may, after the same manner, be demonstrated of the two triangles A B C, A DE, by making the equal angles vertical, as at A, and completing the triangle AC D, that if that is, if AB be to A D the sides be reciprocally proportional, as AE to A C, the triangles ABC, ADE

will have the same ratio to the triangle A CD, and therefore will be and, conversely, that if equal to one another: ABC, ADE be equal

B

to one another, and therefore have the same ratio to the triangle A CD, AB will be to AD as A E to A C, that is, the sides about the equal angles will be be reciprocally proportional; 39. being

cited instead of 35.

PROP. 42. (Euc. vi. 19.) in the duplicate ratio of their homoloSimilar triangles are to one another gous sides.

Let A B C, DEF be similar triangles, and let the sides B C, EF be homologous; the triangle ABC shall have to the triangle D E F the du

AA

plicate ratio of that which BC has to EF.

angle at E, the triangle A B C is to the Because the angle at B is equal to the triangle D E F in the ratio which is compounded of the ratios of A B to DE,

and of B C to E F (40.). But, because the triangles are similar, A B is to B C as DE to E F, and alternando AB: DE:: B C E F; therefore the ratio which is compounded of the ratios of AB to DE, and of B C to E F, is the duplicate of the ratio of B C to EF (37. Cor. 1.). Therefore the triangle ABC has to the triangle DEF the duplicate ratio of that which B C has to E F.

Otherwise: Take B G a third proportional to B C and E F, and join A G. Then the triangle A B C is to the triangle ABG as BC to BG, that is, (def. 11.) in the duplicate ratio of BC to EF. But because AB is to D E as BC to EF, that is, (12.) as E F to BG, the triangles ABG, DEF have their sides about the equal angles B and E reciprocally proportional: therefore the triangle ABG is equal to the triangle DEF (41.). Therefore the triangle ABC is to the triangle DEF in the duplicate ratio of B C to EF,

Therefore, &c.

the figures, are to one another, each to each, in the same ratio. But similar triangles are to one another in the duplicate ratio of their homologous sides. Therefore the triangles into which the figure ABCDEF is divided, are to the similar triangles into which the figure abcdef is divided, each to each, in the same ratio (37. Cor. 4.) viz. in the duplicate ratio of that which AB has to a b. Therefore the sum of all the former is to the sum of all the latter (23.Cor. 1.) that is, the figure ABCDEF is to the figure abcdef, in the same ratio.

Again, because the sides AB, B C, &c. of the one figure are to the homologous sides ab, bc, &c. of the other figure in the same ratio, the sum of the former is to the sum of the latter in the same ratio; that is, the perimeter of the

one figure is to the perimeter of the other as A B to a b.

Therefore, &c.

Cor. 1. Similar rectilineal figures are to one another as the squares of their homologous sides (37. Cor. 2.).

Cor. 2. (Euc. vi. 22.). If four straight lines be proportionals, any similar rectilineal figures described upon the first and second shall be to one another as any similar rectilineal figures described upon the third and fourth; and conversely (37. Cor. 4.).

PROP. 44. (Euc. vi. 31.)

In a right-angled triangle, if similar rectilineal figures be similarly described upon the hypotenuse and the two sides, the figure upon the hypotenuse shall be equal to the sum of the figures upon

the two sides.

For the figure upon one of the sides is to the similar figure upon the hypotenuse, as the square of that side to the the similar figure upon the other side is square of the hypotenuse (43. Cor.1.), and to the figure upon the hypotenuse as the square of that other side to the square of the hypotenuse-proportions having the same consequents: therefore (25.) the sum of the figures upon the two sides is to the figure upon the hypotenuse as the sum of the squares of the two sides to the square of the hypotenuse, that is (I. 36.), in a ratio of equality. Therefore, &c

SECTION 6.-Of Lines in Harmonical Progression.

Def. 17. Three straight lines are said to be in harmonical progression when the first is to the third as the difference of the first and second to the difference of the second and third.*

Of three lines A, B, C, which are in this progression, B is said to be an harmonical mean between A and C, and

This progression was called harmonical from its having been first noticed (it is said, by Pythagoras) in the lengths of chords which, having the same thickness and tension, produce the sounds of a certain note, its fifth and its octave. These lengths are as 1,, and, of which it is plain that the first is to

the third as the difference of the first and second to the difference of the second and third,

It is observable that if harmonical means be inserted between the numbers above mentioned, lengths will be found among them producing the other notes of the major scale.

"If a musical string C O and its parts D O, E O. FO, GO, AO, BO, c O, be in proportion to one another as the numbers

1, §, †, 4, 4, †, T‰, †, their vibrations will exhibit the system of 8 sounds, which musicians denote by the letters C, D, E, F, G, A, B, c." Smith's Harmonics, Sect. II. Art. 1,

68

C a third harmonical progressional to A and B.

After the same manner, also, three magnitudes of any other kind are said to be in harmonical progression, viz. when the first is to the third as the difference of the first and second to the difference of the second and third; and the terms harmonical mean and third harmonical progressional are applied to

them in the same sense.

18. Any number of straight lines, or other magnitudes, A, B, C, D, &c. are said to be in harmonical progression, when every consecutive (or following) three are in harmonical progression.

the third

and

19. A straight line is said to be harNow is an harmonical mean between 1 and 3, the second of two harmonical means between 1 and, and the first of three harmonica, means be tween 1 and . Again, is the first of two harmonical means between 4 and, and of three harmonical means between In fact, taking the original progression 1,,, and inserting first one harmonical mean between its terms, we get the progression 1,,,, ; secondly, two harmonical means between its terms, 1, 4, †, †, †, TT, ; and thirdly, three harmonical means between its terms, 1, 4, †, TT, }, T3, 4, 75, 7; from which progressions, rejecting such fractions as admit 7, 11, and 13 in the denomi nator, that is, such as have other numbers entering into their terms, besides 2, 3, 5, and their products, those which remain will represent the lengths of strings producing, with the same thickness and tension, the sounds denoted by C, D, E, F, G, A, B, c.

8

The above observation, striking and ingenious as it is, must not, however, lead the student to suppose that the theory of Harmonics has any mysterions connexion with the properties of lines harmonically divided. Why such lengths only as are related by

T

the numbers 2, 3, 5, and their products, produce a gradation of sounds pleasing to the ear as those of the gamut, it is for that theory to explain; but the discovery of these relations, by taking harmonical means, is attributable to the simple property, that the reciprocals of numbers in harmonical progression are in arithmetical progression. Thus the reciprocals of 1, †, †, TT, †, T, 4, 5, +, that is of †, J, TO, TI, TI, TJ, T4, T5, T, are fractions having the common denominator 8, and 9,9, 10, 11, &c. for their numerators, that is, are in arith metical progression. And, generally, if a, b, c be in harmonical progression, i. e. if a:c:: a~b: b~c, nultiplying extremes and means, a b~a ca c~bc, and 1 1 1 1

1 1

dividing by a b c,-~ THAT i. e. ——— с b b a α b c are in arithmetical progression. It follows as a necessary inference, that, if the lengths of the strings which produce harmonious sounds, bear to each other a ratio which can be expressed in whole numbers, however great, they may be made terms in some harmonical series; the singular result which arises from this ratio being expressed in terms involving only the numbers 2, 3, 5, and their products, is that the whole series is obained by the interposition only of two and of three

harmonical means between the note and its fifth

and the fifth and octave; for, being comprised in the series which results from the interposition of three means, that of one mean may be neglected.

1

If AB, A C, AD be harmonical progressionals in the same straight line. DC, DB, DA shall likewise be harmonical progressionals. (See figure of def. 19.).

Because AB, AC, AD are in harmonical progression, (def. 17.) AB: AD :: BC: CD; therefore, alternando, AB B C :: DA: CD, and, invertendo, DC DA :: BCA B. Therefore D C, DB, DA are three straight lines, such, that the first is to the third as the difference of the first and second to the difference of the second and third; that is, (def. 17.) DC, DB, DA are in harmonical progression.

Therefore, &c.

Cor. If a given line A C be divided in any ratio in the point B, and if A C produced be divided in the same ratio in the point D (so that DA may be to DC as A B to B C), the whole line A D will be harmonically divided in the points B and C. For it is obvious that DA is harmonically divided in the points B and C; that is, that DA, DB, DC are harmonical progressionals: therefore, also, AD, A C, A B are harmonical progressionals, and AD is divided harmonically in the points B and C.

PROP. 46.

If A B, A C, AD be harmonical progressionals in the same straight line, and if the mean A C be bisected in K, KB, KC, K D shall be in geometrical progression: and conversely.

straight line ab shall likewise be divided harmonically.

Through C draw E F parallel to PA, and let it cut PB, PD in the points E, F respectively. Then, by similar triangles BPA, BEC, (31.) the ratio of AP to EC is the same with that of AB to B C. Again, by similar triangles DPA, DF C, the ratio of AP to CF is the same with the ratio of AD to D C. But, because AD is harmonically divided, AB has to BC the same ratio as AD to D C, (def. 19.): therefore, (12.) AP has to EC the same ratio as AP to C F, and (11.Cor.1.) EC is equal to CF.

And, because E F, which is parallel to PA, has equal parts of it intercepted by PB, PC, PD, if through the point d in which ab cuts PD, the straight line ef be drawn parallel to PB and terminated by PC, PA, the line ef will likewise be divided equally in d(48.Cor.). But, by similar triangles cPb, ced, cb:cd::Pb: ed, and by similar triangles a Pb, afd, abad: Pb: df (or ed): therefore cb: cd::ab: ad,

D

by the straight line AD which cuts the base BC in D: BD shall be to DC as BA to A C.

Through C draw CE parallel to AD, and let it meet B A produced in E: then, because the angles A E C, ACE are (I. 15.) equal, respectively, to the halves of the bisected angle, they are equal to one another: wherefore AC is equal to A E, (I. 5.). But, again, because CE is parallel to DA, BD: DC :: BA: AE, (29.): therefore BD :DC::BA: A C.

And, conversely, if B D be to D C as BA to A C, A D shall bisect the vertical angle. For, C E being drawn (as before) parallel to A D, because B A is to A C as BD to D C, that is (because CE is parallel to D A) as BA to AE (29.), AC is (11. Cor. 1.) equal to A E. Therefore the angle A E Cis equal to ACE (I. 6.), and the parts of the angle in question being equal to AEC, ACE respectively (I. 15.), are equal to