Otherwise: Take B G a third proportional to B C and E F, and join A G. Then the triangle ABC is to the triangle ABG as BC to BG, that is, (def. 11.) in the duplicate ratio of B C to EF. But because AB is to DE as BC to EF, that is, (12.) as EF to BG, the triangles ABG, DEF have their sides about the equal angles B and E reciprocally proportional: therefore the triangle A B G 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. Cor. Since the duplicate ratio of two straight lines is the same with the ratio of their squares (37. Cor. 2.) it appears that similar triangles are to one another as the squares of their homologous sides. PROP. 43. (Euc. vi. 20.) Similar rectilineal figures are to one another in the duplicate ratio of their homologous sides; and their perimeters are as those sides. For it has been seen (32. Cor. 2.) that any two similar rectilineal figures A B CDEF, abcdef, may be divided into the same number of similar triangles by straight lines drawn from corresponding angles A, a; and the homologous sides of these triangles, which are the same with the homologous sides of F A B 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. one figure is to the perimeter of the 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 square of the hypotenuse (43. Cor.1.), and the similar figure upon the other side is 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 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, 2, 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 Again, because the sides A B, B C, &c. of the one figure are to the homologous sides a b, 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 A, B, c." Smith's Harmonics, Sect. II. Art. 1. other notes of the major scale. "If a musical string C O and its parts D O, E 0, FO, GO, A O, BO, c O, be in proportion to one another as the numbers 8 1, 8, 4, 4, 3, 3, T5, 4, their vibrations will exhibit the system of 8 sounds, which musicians denote by the letters C, D, E, F, G, 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. 19. A straight line is said to be harNow is an harmonical mean between 1 and, the second of two harmonical means between 1 and, and the first of three harmonical means be tween 1 and 3. Again, is the first of two harmonical means between and, and the third 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, 4, 3, 3, Tr, ; and thirdly, three harmonical means between its terms, 1, 8, 4, TT, 3, 13, 7, 75, 7; from which progressions, rejecting such fractions as admit 7, 11, and 13 in the denominator, 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. The above observation, striking and ingenious as it is, must not, however, lead the student to suppose that the theory of Harmonics has any mysterious connexion with the properties of lines harmonically divided. Why such lengths only as are related by 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, §, 3, TT, †, 1°5, †, 1°5, 4, that is of 8 , &, Tσ, TI, 17, 15, 14, 15, 15, are fractions having the common denominator 8, and 8, 9, 10, 11, &c. for their numerators, that is, are in arithmetical 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 1 dividing by a bc,-~ ~i. e. --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 c b 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. It will be shown in Prop. 45., that, if the line AD be harmonically divided in the points B, C, the line DA will likewise be harmonically divided in the same points: that is, that if A B, A C, and AD be harmonical progressionals, DC, DB, and DA shall likewise be harmonical progressionals. 20. Four straight lines are said to be harmonicals, when they pass through the same point, and divide any one straight line harmonically. PROP. 45. 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 DC, 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.) D C, D B, DA are in harmonical progression. Therefore, &c. in any ratio in the point B, and if A C Cor. If a given line A C be divided 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, D B, D C 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, K B, K C, K D shall be in geometrical progression: and conversely. In the first place, because A B : AD ::BC: CD, and that AD is greater than CD, AB is also greater than BC, (18.Cor.). Wherefore the point K,which bisects A C, Л B D lies between A and B. Again, alternando, AB: BC: AD: CD: therefore, by sum and difference, A B~B C : AB+BC: AD CD: AD+CD. But the first term of this proportion is equal to 2K B, the second to 2 KC, the third likewise to 2 K C, and the fourth to 2 K D. Therefore 2 KB 2 KC 2 KC: 2 K D; and hence, (17. Cór. 2.) KB: KC:: KC: KD, that is, KB, KC, KD are in geometrical progression. Next, let K B, K C, KD be in geometrical progression, and let KA be taken equal to KC: AB, AC, AD shall be in harmonical progression. For since KB: KC KC: KD, by sum and difference KB+KC: K B~ KC::KC+KD: KC~KD, that is, AB: BC:: AD: CD. Therefore, alternando, A B: AD::BC: CD, or AB, A C, AD are in harmonical progression. Therefore, &c. PROP. 47. The same being supposed, DA, D K, DB, D C shall be proportionals. Because DA is equal to the sum, and D C to the difference of DK, KC, the rectangle under DA, D C is equal to the difference of the squares of DK, KC (I. 34.). Again, because KC is a mean proportional between KB and KD, the square of K C is equal to the rectangle under K B, K D, (38. Cor. 1.). Therefore the rectangle under DA, DC is equal to the difference of the square of D K, and the rectangle under KB, KD; that is, (I. 30. Cor.) to the rectangle D K, DB. Therefore (38.) DA, DK, DB, DC are proportionals. Therefore, &c. Cor. 1. If K B, KC, KD be proportionals in the same straight line, and if KA be taken in the opposite direction equal to the mean KC; DA, DK, DB and D C shall be proportionals (46.). Cor. 2. From this proposition it ap. pears that the harmonical mean D'B between two straight lines DA and DC is a third. proportional to the arithmetical mean D K, and the geometrical mean M between the same two. For DBXDK=DAXDC=M2 if M be a geometrical mean between DA and DC: and because DB × DK=M2, DK, M, and D B are proportionals. PROP. 48. If four straight lines pass through the same point; to whichsoever of the tercepted by the other three, shall be to four a parallel be drawn, its inparts one another in the same ratio. PC, PD pass through the same point Let the four straight lines PA, PB, P; through A, any point in PA, draw AC parallel to PD, and let it be divided by the other three PA, PB, PC into the parts Ab, bc; through c draw Bd parallel to P A, and let it be divided by the other three into the B parts Bc, cd; through d draw Ca parallel to PB, and let it be divided by the other three into the parts Cd, da; lastly, through a draw Db' parallel to PC, and let it be divided by the other three into the parts Da, a b': then, Ab shall be to bc, as cd to Bc, as C d to da, and as a b' to D a. Because Ac is parallel to PD, and cd to PA, Ad is a parallelogram. therefore (I. 22.) Pd is equal to A c. And, by similar triangles B Pd, Bbc, Pd: bc: Bd: Bc, (31.): but Pd is equal to A c; therefore, dividendo, A b bc cd: B c. In the same manner it may be shown that cd: Bc: C d: da; and again, that Cd da :: ab': Da. Therefore the ratio of Ab to bc is the same with the ratio of cd to B c, which is the same again with that of Cd to da, which is the same with that of a b' to Da. And any straight lines parallel to these (30.) will be divided in the same ratio. Therefore, &c. It will be observed that, if the parts be considered as proceeding in a particular direction, viz. from A towards B, C, D, the proportional parts are continually in an inverted order: thus, Ab, is to be not as Be to ed, but as cd to (12.) that is, a b is harmonically divided B c, and so on. in the points c, d. Therefore, &c. Cor. If four straight lines pass through the same point, and if a parallel to one of them has equal parts of it intercepted by the other three, a parallel to any of the others shall likewise have equal parts of it intercepted by the other three. Cor. 1. If there be four straight lines harmonicals (def. 20), and if a parallel be drawn to any one of them, equal parts of the parallel shall be intercepted by the other three. Cor. 2. And, conversely, if four straight lines pass through the same point, and if, a parallel being drawn to any one of them, equal parts of the parallel be intercepted by the other three, those four straight lines shall be har monicals. Thus, the two sides of a triangle, a line drawn from the vertex to the bisection of the base, and a line drawn through the vertex parallel to the base, are harmonicals. PROP. 50. (EUc. vi. 3. & A.) If the vertical and exterior-vertical angles of a triangle be bisected by straight lines which cut the base and the base produced, the base and likewise the base produced shall be divided in the ratio of the sides: and conversely. Let ABC be a triangle; and first, let the vertical angle BAC be bisected D 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 A B to BC. Again, by similar triangles DPA, D F 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 C F. 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, 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 AE C, A CE are (I. 15.) equal, respectively, to the halves of the bisected angle, they are equal to one another: wherefore because CE is parallel to D A, B D: A C is equal to AE, (I. 5.). But, again, DC::BA: AE, (29.): therefore B D :DC::BA: A C. B A to A C, A D shall bisect the verAnd, conversely, if B D be to D C as tical angle. For, C E being drawn (as before) parallel to A D, because B A is to AC as BD to D C, that is (because (29.), AC is (11. Cor. 1.) equal to A E. CE is parallel to D A) as BA to AE 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 When the sides AB, AC are equal to one another, A D bisects the base B C at right angles (I. 6. Cor. 3.); and Ad is parallel to the base BC; for Ad is always at right angles to AD, because the angle D A d is equal to the halves of the two angles BAC, CA E together, that is, to the half of two right angles. Cor. Since the ratios of B D to DC, and of Bd to d C, are each of them the same with the ratio of B A to A C, they are the same with one another (12.), and Bd is harmonically divided in the points D, C. Therefore the two sides of a triangle, and the lines which bisect the vertical and exterior-vertical angles, are harmonicals (def. 20.). SECTION 7.-Problems. Def. 21. A straight line is said to be divided in extreme and mean ratio, when the whole line is to the greater segment as the greater segment is to the less. A straight line so divided is also said to be divided medially; and the ratio of its segments is called the medial ratio. PROP. 51. Prob. 1. (Euc. vi. 13.) To find a mean proportional between two given straight lines A B and B C. Let the straight lines A B, BC be placed in the same straight line: from the point B (I. 44.) draw B D at right angles to AC: bisect A C in E (I. 43.), and from the centre E, with the radius E C, describe a circle cutting BD in D: BD shall be the mean proportional required. A E B For, the angle EBD being a right angle, the square of B D is (I.36.Cor. 1.) equal to the difference of the squares of EB, ED. But, because E A and E C are, each of them, equal to ED, AB is equal to the sum, and B C to the difference of ED, EB: therefore (I. 34.) the square of BD is equal to the rectangle under AB, B C, and (38. Cor. 1.) BD (1.48.), to meet AE produced in F. D Then, because (29.) A B is to B C as AE (or D) to EF, EF is the fourth proportional required. Therefore, &c. PROP. 54. Prob. 4. (Euc. vi. 10.) To divide a given straight line A, similarly to a given divided straight line BC. Method 1. Draw B D making any angle with B C, and make BD equal to A join C D, and through the several points in which BC is divided, draw lines parallel to CD (1.48.): then B A (29. Cor.), because these lines are parallel to CD, the straight line D B, that is, A, is divided by them similarly to the given divided straight line B C. Method 2. Upon BC describe (1.42.) the equilateral triangle DB C: take DE, DF each of them equal to A, and join EF; and from D through the several points in which B C is divided, draw straight lines cutting EF. E |