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64.

PROPORTIONAL LINES AND AREAS.

DEFINITION. Ratio is the relation which two or more things, or quantities of things, of the same kind bear to each other in respect of magnitude. And, for the purpose of this comparison, any two things are of the same kind only when the lesser of the two by multiplication can be made to exceed the other.

Thus a lineal foot can be multiplied (22) until it exceed a lineal mile; therefore these are things of the same kind, and bear a certain ratio to each other. So likewise an oz. and a lb. in weight have a certain ratio; a quart and a gallon have a certain ratio; and so on.

But an oz. and a mile are not things of the same kind. The one can never by multiplication be made to exceed the other; and consequently they bear no relation to each other in respect of magnitude, that is, they can have no ratio to each other.

Similarly, a line may have ratio to a line, and an area to an area; but a line can have no ratio to an area, because by the multiplication of either we can never arrive at, or exceed, the other.

65. DEF. The measure of the ratio between any two magnitudes is, (not their difference, but) the number of times the one contains, or is contained in, the other.

Thus, if the line AB, upon being multiplied three times (22), becomes equal to the line CD, that is, if CD contains AB exactly three times, then the measure of the ratio of CD to AB is 3, that is, CD bears the same relation to AB in magnitude which 3 does to 1.

But in order that two magnitudes of the same kind may have a ratio to each other, it is not necessary that one should contain the other an exact integral number of times.

Thus, for example, let A be a magnitude which contains another magnitude taken as the unit of measurement, whatever that may be, 5 times; and let B be another magnitude, of the same kind, containing the same unit 3 times; then the ratio of A to B will be that of 5 to 3. In this case A may be said to contain B once and two-thirds of a time; and the measure of the ratio of A to B is 13, or the fraction §. Similarly in other cases.

In the case here supposed, a certain multiple of A is equal to another certain multiple of B, that is, three times

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A five times B. Thus, if A be a line which contains a lineal foot 5 times, and B another line which contains it 3 times, then A = 5 feet, and B = 3 feet; the ratio of A to B is that of 5 feet to 3 feet, that is, 5 to 3; and 3 times A 15 feet 5 times B*.

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66. DEF. PROPORTION is the equality of ratios. Thus, if the ratio of A to B be equal to the ratio of C to D, then A, B, C, D are said to be proportionals, or in proportion.

Observe, A, B, C, D, in order to be proportionals, need not be all of the same kind. It is only necessary that A and B be of the same kind, and likewise C and D of the same kind; but the one pair of magnitudes may be different from the other pair. Thus, one line, A, may have the same ratio to another line, B, that one area, C, has to another area, D, in which case A, B, C, D are proportionals.

The ratio of two magnitudes is often expressed by placing the symbol: between them; thus A: B signifies the ratio of A to B. So then, if A, B, C, D are proportionals, A: BC: D; but this is generally written thus, A: B :: C: D; and is read 'A is to B as C to D', which means that A has the same ratio to B which C has to D.

67. PROP. I. If two straight lines be intersected by any number of parallel lines, so that the parts of one of them intercepted between the parallels are equal to one another, the parts also, of the other line between the same parallels shall be equal to one another.

Let ABCD be any straight line, such that AB=BC = CD; or similarly, whatever the number of parts may be of which it is composed. Through the points A, B, C draw parallel lines Aat, Bb, Cc, Dd, meeting another straight line in the points a, b, c, d; then also ab = bc = cd.

Through the point a draw ae D parallel to AB, meeting Bb in e; and through b draw bf parallel

C

B

d

* In this section single letters will often be used to denote lines and other magnitudes, to avoid superfluous writing, where it may be done without risk of error.

+ This is read' A little a,' 'B little b', &c.

to BC meeting Cc in f. Then since A Bea, and BCfb are parallelograms, ae = AB, and bf=BC (40); but AB =BC, :. ae=bf.

Again, because Bb is parallel to Cc, LA Be = 4 BCf; and because ae is parallel to AB, LABe: = 4aeb; also because bf is parallel to BC, BCf=4bfc, .. 4aebbfc. And since ae, bf are parallel to the same line ABC, they are parallel to each other, and ... < eab4fbc. Hence in the two triangles aeb, bfc, there are two angles in the one equal to two angles in the other, each to each, and the side common to those angles in the one equal to the side which is common to the two angles, equal to them, in the other, .. the triangles are equal in all respects, and the side ab the side bc (39). Similarly it may be shewn, by drawing cg parallel to CD, that bc=cd; .. ab=bc=cd. And the proof may be extended to any any number of parts.

COR. 1. The proof here given is independent of the length of the line A a, and will therefore hold when A and a coincide in the same point, that is, when the two given lines meet in A.

COR. 2. Conversely, if two straight lines be composed of the same number of equal parts, the straight lines Aa, Bb, Cc, &c., joining corresponding points in them, will be parallel.

68. PROP. II. To divide a given straight line into any number of equal parts.

Let AB be the given straight line, which is to be divided into any proposed number of equal parts, three suppose, as the process is the same whatever the number may be.

b.

A

F

From A draw any other indefinite line Ab, forming an angle with AB; in Ab take any point D conveniently near to A, and with centre D and radius DA

describe a circle cutting Ab in E; then with centre E and the same radius as before describe a circle cutting Ab in C; so that ADDE = EC. Join CB, and through E and D draw EG and DF parallel to BC, cutting AB in G and F. Then since AD = DE = EC, and CB, EG, DF are parallels, .. also AF = FG = GB, that is, AB is divided into three equal parts in the points F and G, (67 Cor. 1.)

69. PROP. III. If any two straight lines be cut by three parallel straight lines, the parts intercepted between the parallels shall be proportionals'.

Let ABC, DEF be any two straight lines, and AD, BE, CF, three parallel straight lines intersecting the former in A, B, C, and D, E, F. Then AB, BC, DE, EF shall be proportionals', that is, AB shall have the same ratio to BC that DE has to EF.

Let AB contain the line which is the unit of measurement three times, and BC the same unit five

B

b

times, (the same proof will hold for any numbers), and divide AB into three equal parts, and BC into five (68), and through the points of division draw lines parallel to BE intersecting DE, and EF: then DE will be divided by these parallels into the same number of equal parts as AB, and EF into the same number as BC (67), that is, DE will contain a certain line three times, and EF the same line five times; or the ratio of DE to EF is three to five, which is the same ratio as AB to BC; .. AB, BC, DE, EF are proportionals (66).

Observe, it may be that a specified unit of measurement will not exactly divide AB, and BC, in which case the unit must be reduced, until this can take place. For example, if there be not an exact number of feet in AB, and BC, there may be an exact number of inches; or, if not inches, there may be an exact number of tenths of an inch; and so on. And whatever be the reduced unit which will exactly divide both AB and BC, the proof above given then holds.

70. PROP. IV. If two sides of a triangle be intersected by a straight line parallel to the third side, the two sides are divided proportionally.

Taking the preceding fig. in (69), through A draw Abc parallel to DEF, cutting BE in b, and CF in c. Then ACc will represent any triangle having its two sides AC, Ac, intersected in B, b, by Bb which is parallel to Cc; and it is required to prove that AB is to BC as Ab is to bc.

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Since ADEb is a parallelogram, Ab = DE. Similarly be EF; and, by (69), it is proved that AB is to BC as DE is to EF, .. AB is to BC as Ab is to bc; that is, AC, Ac are divided proportionally in B, b.

COR. It follows that AB is to AC as Ab is to Ac.

For since AB is to BC as Ab is to bc, this means that AB contains, or is contained in, BC, the same number of times that Ab contains, or is contained in, bc. Now it is plain that, whatever be the number of times AB contains, or is contained in, BC, it is contained in, AB+BC, or AC, exactly once more. Also whatever be the number of times Ab contains, or is contained in, bc, it is contained in, Ab+bc, or Ac, exactly once more. But, if each of two equal numbers be increased by 1, they will remain equal; .. AB is contained in AC the same number of times that Ab is in Ac; that is, the ratio of AB to AC is equal to the ratio of Ab to Ac.

The common mode of writing the two last results is,
AB: BC :: Ab: bc, and AB: AC :: Ab: A c.

71. PROP. V. Similar triangles have the sides forming the equal angles proportionals.

[DEF. Similar triangles are such as have their angles equal, each to each.]

Let ABC, abc, be similar triangles, that is, A Lα,

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