with which the prevailing opinion has so long charged it.

How far these objects are attained remains for the impartial scientific public to decide, on whose judgment alone depends its approval or condemnation.

COLLEGE FOR CIVIL ENGINEERS,

Putney, 19 November, 1840.

BOOK V.

DEFINITIONS.

I.

A LESS magnitude is said to be an aliquot part or submultiple of a greater magnitude, when the less measures the greater; that is, when the less is contained a certain number of times exactly in the greater.

Thus, if

ᄆᄆ ᄆᄆᄆ

be contained in

three times exactly, the former is said to be a submultiple of the latter. The number 5 is said to be an aliquot part or submultiple of 20, because 5 measures 20: i.e., 5 is contained in 20 four times exactly; the same may be said with respect to other numbers and magnitudes.

II.

A greater magnitude is said to be a multiple of a less, when the greater is measured by the less; that is, when the greater contains the less a certain number of times exactly.

Let the line A be 6 feet, and B one foot in length;

A is said to be a multiple of B, because A contains B a

A

B

certain number of times exactly. Again; 24 is said to be a multiple of 2, because 24 contains 2 twelve times, without leaving a remainder.

B

III.

Ratio is the relation which one quantity bears to another of the same kind, with respect to magnitude.

Several definitions have been given of the term "ratio;" the following are quoted from some of the best writers on proportion:—

"Ratio is a mutual relation of two magnitudes of the same kind to one another, in respect of quantity."-Simson's Euclid.

"Ratio is the relation which one quantity bears to another in respect of magnitude, the comparison being made by considering what multiple part or parts one is of the other."-Wood's Algebra.

"Ratio is that relation of two quantities of the same kind, which arises from considering what part or parts the one is of the other.”—Bonnycastle's Geometry.

"Ratio is usually defined to be the relation which one quantity bears to another of the same kind, with respect to magnitude: and as such relation may be expressed either by stating how much one exceeds the other, or how often one contains the other; ratio has accordingly been divided into two kinds-arithmetical ratio, and geometrical ratio.

"Arithmetical ratio is that which expresses the difference of the quantities compared.

"Geometrical ratio expresses the quotient arising from the division of the quantities compared."-Young's Algebra.

If there be two lines, one of which is 16 feet long, the other 3, the length of the former is said to have to that of the latter the ratio of 16 to 3; generally written, 16 : 3, in which the first term (16) is called the antecedent, and the other (3) the consequent. Or the ratio is sometimes written in the form of a fraction, thus, equal 53, showing that the antecedent contains the consequent 5 times.

The area of a parallelogram, whose length is 4 feet and breadth 3, is to the area of another, whose length is 8 feet and breadth 6, in the ratio of 12: 48; and may be expressed by the fraction = 4, showing that the antecedent is of the consequent.

Let there be two cubes, each of the linear edges of the first is 4 feet, and of the second 2 feet; then the length of all the linear edges of the first is to the length of all the linear edges of the second, as 48: 24: the area of the whole surface of the first is to that of the second, as 96: 24, and the solidity of the first cube is to that of the second in the ratio of 64 to 8. In common language these ratios would be expressed by saying, the first cube is eight times as big

as the second, the surface of the first is three times as great as the surface of the second, and the linear sides of the first are twice as long as those of the second. But, if the first of these be a cube of water, and the other copper, the ratio of the weight of the first is to that of the second very nearly as 8:9, or the first is only ths the weight of the second, although it is 8 times as big; this can be easily shown, as a cubic foot of water weighs 1,000 ounces, and that of copper 9,000 ounces nearly.

The term ratio is used in comparing different degrees of heat and light, and other things quite foreign to geometrical magnitudes. The doctrine of ratios, generally treated, requires not the aid of numbers, but the moment we descend to particular cases the idea of number presents itself; and, in many cases, numbers are inadequate to express exactly the ratios of geometrical magnitudes, or even the relations which exist among one another; and yet although the ratios referred to cannot be expressed exactly by numbers, they can be expressed to any designed degree of exactness; in such cases the term "ratio nearly" is applied. Of this we will give one or two instances here:When the diameter of a circle is 1, the circumference is 3.14159265 very nearly; for although we cannot find what the true circumference is, yet we know that 3.14159265 does differ from it 1000000。 part of a unit: from this we infer that the ratio of the diameter to the circumference is as 1 : 3.14159265 nearly. The ratio of the square root of 2, to the square root of 7, is 1.4142136: 2.6457513 nearly. 1.4142136 is not the exact square root of 2, nor can the exact square root be obtained; but yet we may approach to it to any designed degree of exactness. The number above given does not differ 10000000 part of a unit from the square root of 2, the same may be said of the number 7; therefore we infer that 1.4142136 2.6457513 expresses the "ratio nearly" of the square root of 2 to the square root of 7. The term ratio has been applied by mathematical writers to signify different relations, besides that relation which Euclid intended it to express; this has led to a great deal of confusion, and should be discontinued, or the difference shown when such term is used. Some writers differ so far from Euclid's plan, as to say, "it matters not whether we consider how often the first term contains the second, or how often the second contains the first;" now, according to the principles laid down in Euclid's Fifth Book, 12 : 3 is said to be a greater ratio than 12: 4, because 12 contains 3 a greater number of times than 12 contains 4. Quite the contrary conclusion must be come to, if we consider how often the second term contains the first. This latter plan of comparing ratios must be instituted for the purpose of differing from Euclid, as it is not in any way superior; and besides, the disorder that must follow in the comparison of ratios, by plans so widely differing; for that which is called greater ratio by one, is a less ratio by the other.

B 2