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pairs there is the same ratio, are said to be proportioned to each other, or simply to be proportionals. Any two pairs of either set of those above shown are proportionals, and may be written thus, % or 2: 4: 5: 10, as explained in page 703, and 2 the common enunciation in words is, "As the number 2 is to 4, so is the number 5 to 10."

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$ 65. Every schoolboy is familiar with this form, as being that called the rule of three, and he knows that when the three first are placed in order, as—

3:4: 12:

if he multiply together the second and third numbers, and divide the product of these by the first, he obtains as quotient a fourth number, having the same relation to the third that the second has to the first. This is the process,

34: 12: [16]

4

3)48(1

3

3)18(6

18

0

The new number here found is enclosed in brackets. One sees that the fraction is equal to 1 (

1).

§ 66. Simple as this arithmetical operation is, the reason is not at first clearly perceived why the answer or result obtained by the rule must always be rigidly correct. The explanation is as follows.

§ 67. On considering the case here given, of 3 : 4 :: 12:16, it is evident that the second term 4 is greater than the first, 3, by exactly a third part of the first; for that part, when added to the first, makes it equal to the second. If, therefore, a third part of the third term 12 (viz. 4) be added to it, making the sum 16, that sum will have the same relation to 12 that 4 has to 3.

68. Another view of the case is this. As 3 is to 4, so must 12 times 3 be to 12 times 4, for this is merely multiplying the two terms of a fraction by the same number (see § 33). Now 12 times 3 being 36, and 12 times 4 being 48, the fraction & must be equal to , as thus stated (x = 6). And then to, 1 the two terms of the new equivalent fraction being divided by

any one number, will produce still another fraction of the same

(3)36(12

value. Therefore, dividing 36 and 48 by 3 (3)18(16)

will leave 12 and 16, having to each other the same ratio as 3 to 4. The fraction is therefore proportional to, or 3: 4 :: 12 : 16. This view may be stated in fewer words. If both terms of the fraction be multiplied by 12, an equivalent fraction 3§ is produced, and if both terms of this new fraction be divided by 3, another new equivalent fraction, 1, is produced, by which the question is solved. The rule of three, therefore, which directs to multiply together the second and third of any three numbers following one another, and to divide the product by the first, offers as the quotient a new number, having the same ratio or proportion to the third which the second bears to the first.

§ 69. It is often convenient to substitute letters for the numbers in stating the terms of a proportion, thus

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and the mode of working the rule of three may then be thus

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§ 70. The important fact has to be noticed here, that of four such proportionals as above described, the two middle or mean terms multiplied together, give the same product as the two extremes so multiplied. Thus, in the example above given,

34 12: 16
a: b :: c : d

3 multiplied by 16 give 48, and 4 by 12 give 48, or a d = b c. Obviously, if b be the tenth (or other) part of a, and d, the tenth (or same other part) of c, the whole of c multiplied by the tenth of b must be equal to the whole of a multiplied by the tenth of c.

$71. FORMULÆ,

Having now explained generally the nature of the great fundamental operations of arithmetic, there remain to be considered the formulæ, often in the form of equations, which guide us in ap plying these operations in the computations made for practical purposes.

§ 72. Previously we shall note that besides the sign of the oblique cross (×) used for multiplication, as explained in page 707, the same purpose is served where letters are used to represent numbers, by simply placing the letters together as they stand in printed words. Thus if a be used for 5 and 6 for 4, ab signifies 4 times 5, or 20, and not, as might be supposed, the mere addition of 5 to 4, making the sum of 9.-And with respect to the sign of equality (=), it is to be noted that it signifies, the equality not of the two letters or numbers placed closely on opposite sides of it, but of the whole compound expressions or series of numbers so placed. Such formulæ, called equations, are of singular power and importance in facilitating complex computations. They may be compared to a weigh-beam with the two scales balanced while charged with a variety of known weights, among which some unknown are at first mixed. The amount of the unknown is gradually discovered by shifting from one side to the other, or wholly withdrawing, equal known parts without disturbing the equipoise, until at last only what is known is left in one scale and what was unknown in the other, now ascertained by the known counterpoise.

§ 73. General Proportion. When there are two particulars having such mutual relation in amount, that if one of them be increased or diminished in any degree, the other is changed in the same degree—as the weight of a moving body and its quantity of matter, the two are said to be directly proportioned to each other; and if letters are used to indicate the particulars, as W for weight and Q for quantity of matter, these in writing are connected by placing between them the sign of proportion, two dots (:) thus, W: Q, or the sign of equality (=) thus, W = Q. The use of such proportional expressions is to enable certain new results to be deduced from certain others already accurately ascertained by experiment.

§ 74. Now with respect to motion and rest among bodies, universal observation has shown to be true what is stated in this work (see the Analysis, page 33), that "to change any present state of motion or rest in a body, force proportioned to the change is required, whether to give motion, to take it away, or to bend it." This is a general expression for what have been called the laws of motion.

§ 75. To express shortly a large number of the facts, such initial letters as the following are employed :

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§ 76. To compare given amounts of all such particulars, it is necessary to choose standard units of comparison that can be numbered—as ounces or pounds for quantity of matter, feet or inches for length or distance, seconds or hours for time; for to talk of dividing space by time, or multiplying weight by velocity, has evidently no definite meaning.

In pages 53 and 54 are given many examples, showing relations among Quantity of Matter, Force, Velocity, Momentum, &c. An arithmetical expression for momentum is M = Q x V or M: Q V, signifying that momentum is proportioned to the quantity of matter and the velocity conjointly.

§ 77. In respect to Velocity, V, the degree depends on the relation between the space passed through and the time elapsed during the passage. If, of two bodies, one move three feet in a second, and the other two feet in the same time, the velocity of the first to that of the second is as 3 to 2. In equal times, the velocities are as the spaces, V: S; but if the times be unequal, the velocities are as the units of space divided by the units of time. Thus if one body pass over 10 feet in 2 seconds, and another pass over 8 feet in 4 seconds, the velocity of the first is to that of the second as 10 to or as 5 to 2; for, by dividing 10 feet by 2 seconds, the speed of the body is seen to be 5 feet in one second, and by dividing 8 feet by 4 seconds the speed of the other body is seen to be 2 feet in one second, and their respective velocities therefore are as 5 to 2. This fact is expressed by the formula V = the velocity is as the units of space

S

T

divided by units of the time. The velocity of motion, therefore, through a given space in a given time, is greater just as the time elapsed is less, and is expressed by the quotient of the units of space divided by the units of time.

§ 78. From the formula M: Q V, meaning that the momentum

or quantity of motion in a body is proportioned to the quantity of matter multiplied by the velocity, it may be deduced that a body of 10 pounds weight, moving with a velocity of 10 feet in a second, has the same momentum as a body of 5 pounds moving at the rate of 20 feet in a second; for 10 multiplied by 10 make a hundred, and 5 multiplied by 20 make a hundred (10 × 10 100, and 5 x 20 = 100). (Art. 139.)

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§ 79. That the quantity of motion in a body is proportioned to the force producing it (M = F), is shown in the fact, that the velocity of a falling body increases in exact proportion to the time during which the force of its weight or gravitation has been allowed to act upon it.

§ 80. From pages 101 to 105, the very important fact or principle is described in detail, that a small weight attached to the arm of a lever, or weigh-beam, at a given distance from the axis or fulcrum, has force to balance a greater weight or resistance acting on the other arm at a shorter distance from the fulcrum; and that, for exact balance, the smaller weight must be as much further from the fulcrum than the other, as the other is weightier than it.* If the small weight be 2 pounds and its distance 8 feet, and the greater weight be 4 pounds, the distance of that must be 4 feet. The letter P may indicate the small weight or power, R the greater weight or resistance, p the distance of the small weight, and r the distance of the larger. Then the units of weight and distance P, p, multiplied together must always give the same product as the units of R r. The facts thus expressed have the form of what is called an equation, P p = Rr; and any three of the particulars being given, the remaining one may be found by working the rule of three in regard to it, thus:

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§ 81. Such relations among quantities as here spoken of, are interestingly exemplified in the case of a body set free to obey the force of gravitation during its fall to the ground. This phenomenon is fully analysed in the pages from 61 to 65, and is reviewed in page 716. As these supplementary pages are printed separately, the illustrative diagram is here repeated.

* At page 103, line 18, the letter c is to be placed instead of b.

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