245 IV. INVOLUTION AND EVOLUTION. 17. The first of these operations, being merely that of Multiplication, is mentioned here only because the character and circumstances of the direct Arithmetical process constitute a necessary and essential part of the grounds upon which we must endeavour to perform the inverse operation of Evolution. Since the square of 28 = 28 × 28784, the square root of 784 must be 28: and we have to arrive at the latter of these numbers by means of the former: but as there appears to be no immediate connection between them, we put 28 in the form 20+ 8, and then determine the form of its square, from the consideration that the product of two quantities is the sum of the products which arise from multiplying every part of one of them by every part of the other: thus, 20 + 8 20+ 8 400+ 20 × 8 + 20 × 8+ 64 and the square is 400 + 2 × 20 × 8 + 64 which consists of 400 = 202, together with twice the product of 20 and 8, and 64 = 82: in order therefore to ascertain the square root of 784, expressed in the form 400 + 2 × 20 × 8 + 64, we first find the square root of 400 to be 20: and then from dividing 2 × 20 × 8 by the double of this, or by 2 × 20, the remaining part 8 of the root is obtained: so that 2 × 20 + 8 being now made the divisor, and multiplied by 8, and the product subtracted from 2 × 20 × 8+ 64, it appears that the entire root 20 + 8 or 28, is determined. Keeping in view the demonstration above given, we may have either of the following forms of operation: 784 (20+8 400 2 × 20+8=48) 3 8 4 384 784 (28 4 48) 3 8 4 384 the latter being nearly the same as the former, by omitting the ciphers, as was done in Multiplication and Division: and we observe that the Rule laid down in Article (172) of the text, is here investigated for the particular number under consideration. Since, the square of 49 = = 492 == (48 + 1)2 = 482 + 2 × 48 +1, it is obvious that when the root is increased by 1, the corresponding square is increased by twice that root + 1: and the same mode of reasoning being applicable in every other instance, it follows that the remainder at any stage of the process can in no case exceed the double of the root already obtained: agreeably to the observation made at the end of Ex. 4 of Article (172). The method of Multiplication used in this and some of the preceding Articles of the Appendix will furnish the means of deriving the square of one number from that of another by a very simple proceeding: thus, the the square of 31 = (30 + 1)3 = 900 + 2 × 30 + 1 = 961: square of 53 = (50 +3)3 = 2500 + 2 × 50 × 3+9=2809. 18. The rule for the extraction of the cube root given in Article (174), may be investigated in a similar manner, and the observation at the end of the Article may be established upon the same principles; but for the reason stated in the text, it will not be necessary to follow up the inverse processes further in this place, inasmuch as they are rendered much clearer by the use of Algebraical symbols: and the rules already laid down are quite sufficient for the performance of the operations in every case that can occur. V. RATIO AND PROPORTION. 19. The relation of two magnitudes may be known by considering how much the one is greater or less than the other, or what is their Difference, as well as by observing how many times the one is contained in the other, or what is their Quotient. The former of these views, called Arithmetical Ratio, constitutes the business of the operation of Subtraction; and the latter is termed Geometrical Ratio, because it is applied to Geometrical Magnitudes, though it derives its importance from the various uses that are made of it in the calculations of civilized life. In whichever way the comparison may be made, it is evident that no relation can be established between them unless the magnitudes are of the same kind; and consequently Ratio as used in the text must be an abstract quantity, expressing merely the numerical value of one of the magnitudes, with reference to the other considered as an unit of the same kind. Hence, it follows that the ratio or relation, of any two concrete magnitudes of the same kind, as two sums of Money, may be the same as, or equal to, that of two other concrete magnitudes of the same kind, as two bales of Goods: and this Equality of Ratios has been defined to be a Proportion. 20. It is clearly impossible to institute any arithmetical comparison of Geometrical Magnitudes, without the assistance of their Arithmetical Representatives, which it may not always be in our power accurately to obtain ; and to avoid this difficulty, it is stated in the fifth Book of Euclid's Elements, that "Proportion is the Similitude of Ratios; and the first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any equimultiples whatever of the first and third being taken, and any equimultiples whatever of the second and fourth; if the multiple of the first be greater than that of the second, the multiple of the third is also greater than that of the fourth; if equal, equal; and if less, less." be This conclusion has been established in the text with respect to numbers forming a proportion, and it may applied immediately to shew whether four numbers taken in order constitute a proportion or not. Thus, if 2:34:5; by taking equimultiples of the first and third, we have 12 3 24: 5; and by taking equimultiples of the second and fourth, we obtain 12 12 24 : 20, in which the condition above enunciated not being fulfilled, we are assured that the numbers 2, 3, 4, 5 do not form a proportion according to the geometrical definition, as the arithmetical definition shews at once, because is not equal to . VI. APPROXIMATE DIMENSIONS, AND CONTRACTED DECIMAL OPERATIONS. 21. In the Instrumental measurement of Dimensions, we can seldom be assured of absolute accuracy; and, in practice, we are therefore obliged to be content with approximations to their true values: but, though it might be impossible to obtain numerical representatives which are perfectly correct, still there is no difficulty in assigning the limits within which the errors shall be confined. Suppose a measured distance to have been found to be 10.7459 yards, in which the number of entire yards is 10, with the parts of a yard expressed by the decimal .7459: since this magnitude differs from 10 yards by .7459 yards in excess, and from 11 yards by .2541 yards in defect, it will be more nearly expressed by 11 than by 10 yards, if the nearest whole number only be taken. If one decimal be used, 10.7 yards will be correct within a tenth part of a yard, for the differences between the true value and the distances 10.7 and 10.8 yards being .0459 and .0541 yards, the error is less than .05, or of a yard in the former, and greater than this quantity in the latter. If two decimals be retained, the distance is more nearly 10.75 than 10.74 yards, the former being .0041 in excess, and the latter .0059 in defect; or, 10.75 is correct within a hundredth part of a yard, the error being less than ,005, or zoo of a yard. Similarly, 10.746 is in excess, but correct within a thousandth part of a yard: and hence, whenever one or more decimals on the right hand are omitted, care must be taken to increase the last remaining figure by 1, when the first of the figures so left out, is equal to, or greater than 5. Thus, the values of 18.3546. miles within a tenth, a hundredth, a thousandth, &c., of a mile, will be respectively 18.4, 18.35, 18.355 &c., miles. 22. The limits of the errors in computations conducted by means of approximate dimensions may hence be ascertained. Thus, if two dimensions be 2.87 and 3.76 inches, which are both correct within an hundredth part of an inch, the corresponding rectangle will be represented by 2.87 × 3.76 10.7912 square inches: now, 2.87 and 3.76 have the superior and inferior limits 2.875, 2.865 and 3.765, 3.755 respectively, so that the area of the rectangle will lie between 2.875 × 3.765 = 10.824375, and 2.865 x 3.755 = 10.758075, of which the former is .033175 in excess, and the latter is .033125 in defect: but if we divide the sum of the multiplier and multiplicand by 200, the quotient is .03315, which differs slightly from the errors just found: and the same kind of reasoning being applicable in all other cases, we conclude generally that when the factors are true to one, two, three, &c., decimal places, the product will be correct within th, 20th, 2000th, &c., parts of the sum of the said factors. These inferences are established by means of general symbols in (4) of Article (84) of the Author's Elements of Algebra, and though not absolutely correct, they may be relied upon in most practical calculations. A similar test may be applied to Division, since the the dividend × the divisor the dividend (the divisor) by quotient = the divisor dividing the sum of the dividend and divisor by 20 times, 200 times, 2000 times, &c., the square of the divisor. By such considerations we are led to the contracted Multiplication and Division of Decimals, which are performed according to the following directions, whose grounds are easily perceived. 23. In Multiplication, place the units' figure of the multiplier immediately under the decimal place of the multiplicand which is to be retained in the product, and write the rest of the figures of the multiplier in a reverse order: then, in multiplying, reject all the figures in the |