and these, involved into the depth, which, also, is two, give eight, the number of half-foot cubes in one cube of a foot.

164. Second. The three square pieces, or parallelopipeds, as they are termed, are each half a foot thick. And being a foot square, they are, each of them, half of a cubic foot. The whole foot gives us eight of our gauge, these pieces, then, contain each four, and the three contain twelve of that gauge.

165. Third come the three prisms, marked c c c, these are each half a foot square, and one foot long, they contain, therefore, each two cubic half feet; that is, six of our denominator in the three. And, though last and least, not to be forgotten, comes our gauge, or denominator itself, the single cube of half a foot.

166. Let us now add these several quantities together. In the foot cube we found 8, in the three parallelopipeds 12, in the three prisms 6, these make 26, to which add the small cube itself, and we have 27 of the denominator we have chosen. Now, our whole number is the foot, the cube foot; this foot, divided into eighths, gives us cubes of half a foot; the cube of a foot and a half contains, as we have found, 27 of these, that is 27 eighths.-Such is the geometrical calculation. And, now, what says the arithmetical process. Turning to paragraph 150, we see that I raised to its third power is 27.

167. Hitherto, however, we have not traced the process of involution in decimal numbers. It is, in itself, so simple an affair, that a single example may suffice, and for this example let us take one of the quantities that we have already treated as a vulgar fraction; let us take and treat it as a decimal.

5

And first as to the

square. Arithmetically the state

16

100

ment and process will stand thus,— = .4; that is
The square of the decimal. 4 is. 16;
much for the arithmetical process.
geometrical.

168. Take a line of any length, a as a b c. Let this line represent the whole or integral number. De- a cimate it, that is, divide it into tenths. Now what can be more simple, or more clear; four of these decimal parts are. And if we square the whole line, and then divide that square into ten equal parts each

4

4

10

that is So Now for the

b

b

с

way, forming thereby ten-times-ten, that is 100 small squares, as is seen in the figure, then will the square of a b, that is, of four tenths of the whole line, be found to contain sixteen of the hundred; that is, according to the arithmetical expression, of the small

squares.

16

169. And, now for the Cube of this decimal fraction. Arithmetically the matter stands thus .4 × .4= .16; and .16 .4.064-see rule in paragraphs 91 and 92. Observe, now, this decimal, stated with its denominator, would be see paragraphs 76 and 77; or, rather, now that we drop the decimal form, it is 64 ; and bear in mind that the quantity involved, when stated in like manner,

1000

064 1000'

is

4

10

170. Let the line A в be 10 inches, and let a c be four of those inches. Now imagine a cube raised

on the whole of this line. Such a cube would con

tain 1000 cubic inches; for 10 x 10 x 10 = 1000. Then imagine the figure itself, to answer the purpose, must be inconveniently large-so imagine a cube raised on the part A c, that is, on 4 inches; such a cube would contain 64 cubic inches. And if considered as being raised within the larger cube, if considered as being formed of a part, that is, as being a fraction of the cube of 1000 inch cubes, then is this cube of 4, which comprises 64 inch cubes, sixtyfour thousandths of the whole cube. And this agrees with the arithmetical expression of the quantity, as stated in the foregoing paragraph. Such, so simple, is the principle of involution of decimal quantities.

171. Having dismissed this subject of Involution, I may be permitted, as it can here be done with great advantage, to say a few words on two other points, one of which has been reserved until now.

172. In paragraph 95 we involved the decimals 4.4.4, and the result was .064. Now, why is this cipher before the two significant figures? It is according to rule laid down and illustrated in paragraphs 91 and 92; but, WHY THIS RULE? We have just seen the WHY. The rule is, that in a multiplication in which decimals are factors, on the conclusion of the process, there shall be the same number of decimal figures in the product as there are in the factors multiplied together, and that, if, as in this case, of the multiplication of three decimal figures, there be one, or indeed any number of figures short, we shall make up such deficiency by prefixing a cipher or ciphers. Now, the REASON, that is to say, THE PRINCIPLE OF THIS RULE is become apparent; .4 ×.4×.4 without this rule, produces.64; that is 100 if we place its denominator under it according to rule, paragraph 77. But we have seen; in fact, we have it demonstrated in the foregoing process of the

64

involution of these three numbers, that the result is not sixty-four hundredths, but only a tenth of that quantity, that is to say, 1000' the decimal expression

64

of which is, and must be, .064.

173. Need another word be said in order to show, in order to prove, the entire dissimilarity between this, which can only, with propriety, be called, the INVOLUTION OF FRACTIONAL NUMBERS, and any process which can be termed multiplication. We have before argued the case, in the 59th and a few of the succeeding paragraphs, but here the matter has become so clear, the foregoing diagrams and reasoning so palpably show that it is NOT A MULTIPLICATION, as the word multiplication is used in any other case whatever, but that it is clearly an arrangement, merely, of those quantities that are expressed by the figures and numbers with which the operation is performed; an arrangement of those quantities in the form of square, or of cube: that it is no augmentation, nor a diminution neither; but a separate squaring, or cubing, of the two terms of the fraction; that it is, in fact, their INVOLUTION merely: that all the incongruous notions about quantities, or numbers, being diminished when multiplied by a fraction, and increased when divided by a fraction, that all these notions ought to be dissipated. And, further, is it become manifest, that this multiplication, by the two terms of a fraction, can be of no use, can never have any meaning, save in a case of Involution; that is to say, save in a case in which the two terms of a fraction are to be involved for the purpose of ascertaining the proportion which the fractional part bears to the whole or integral quantity contemplated in the given case.

OF EVOLUTION;

OR,

THE EXTRACTION OF THE ROOTS OF NUMBERS.

174. In the foregoing lesson we have seen that, to involve, is to roll, to fold up; so, to evolve is to unfold. And as the term Involution is applied to that process of arithmetic by which a number is raised to its several powers, that is, to its second power, or square; to its third, or cube, and so forth; so to evolve is to retrace this process, and thereby to discover the number which has been raised, or which is susceptible of being raised to a given number. And the term EVOLUTION is applied to the process by which such number may be discovered; and the number so discovered, is called THE ROOT of the given number: for example, let the given number be 16, which is the second power, or square of 4; or let it be 64, which is the cube, or third power of the same number. In these cases, 4 is called the root of the given numbers; it is the square root of 16, and the cube root of 64; and the discovery of it is called Evolution; or the Extraction of the Root of those numbers.

OF THE EXTRACTION

OF THE

SQUARE ROOT.

175. The square roots of certain even numbers, as of 100, or of 400, is seen at once. We see that. 10 x 10 = 100, and that 20 × 20 400. Again, the root of 144 is seen at once to be 12; as may be the root of any square number yet smaller than this. But in real calculations, for purposes of business, we

M