tion should still be retained, notwithstanding its evident imperfections, and the superior claims of the other scales.

The number ten has only two aliquot parts; (exclusive of unity) and, therefore, it is not so convenient, for the radix of a system, as the number six; for though the latter has, likewise, but two aliquot parts, yet, since one of these is a common multiple of both radices, and the powers of the remaining aliquot part of six include a greater proportion of number than those of the remaining aliquot part of ten, it will be found that the Senary Scale also includes a greater proportion of finite radical fractions than the Decimal Scale. Ten, however, is preferable to six, both with regard to conciseness of numerical expression and despatch in calculation, and these properties, in some measure, make up for its defects in the fractional arithmetic.

The Duodecimal Scale combines all the advantages of the Senary and Decimal Scales; it is no less convenient than the one, with respect to its aliquot parts, and still more so, than the other, with respect to the brevity of its operations. Nor is the number twelve so great, as to render computation, by the Duodecimal Scale, at all difficult; on the contrary, it seems to have been resorted to, in every age, as the most convenient number for the divisions of weights and measures. I have, therefore, no hesitation in giving it a decided preference to the decimal system. The Duodecimal Scale (says an able writer) would nowhere have been found of greater use, than when applied to the circle, the case in which the decimal division is liable to the strongest objections.

The only scale which, in my opinion, can at all be compared with the Duodecimal Scale, is the Trigesimal; for the number thirty has the aliquot parts of ten and twelve, at least the prime aliquot parts; and, in the present inquiry, we must consider only the prime aliquot parts of the radices, since it is the number of these alone that constitutes the value of a particular scale, in the Fractional Arithmetic. Thus, though four is an aliquot part of twelve, and not of thirty, the Duodecimal Scale has no advantage, on that account, over the Trigesimal; for, four being a power of two, and two an aliquot part of some power of thirty, four must also be an aliquot part of some power of thirty; and, consequently, those fractions, which have four for their denominator, must be finite radical fractions, by the Trigesimal Scale.

The fractions contained in the parentheses are the fractions that are finite, by the scale, to which they belong, when they are reduced to equivalent fractions, whose denominators are some power of the radix, and the column of integers, on the right hand, expresses their number. The column of fractions, on the right hand, points out the number of finite and interminate radical fractions, with every possible numerator, excluding those fractions which occur, under different forms, of the same value; as, 4, 4, 3, &c.; }, }, %, &c.; the numerators express the finite, and the denominators the interminate radical fractions. It is not to be supposed, however, that the column of fractions expresses the real proportions of the finite and interminate radical fractions of each scale; for, in order to obtain these proportions accurately, it would be necessary to extend the denominators of the fractions, in the Table, from two to some number, which is a common multiple of all the radices.

6

It has been already remarked, that the value of a scale, in the Fractional Arithmetic, depends entirely on the number of its prime aliquot parts; and, therefore, no new scale, after thirty, would give any advantage, till we come to two hundred and ten, the product of the prime numbers, 2, 3, 5, 7. The number two hundred and ten is evidently by far too great for forming the radix of an arithmetical system; and, accordingly, we must restrict our choice to the Duodecimal and Trigesimal Scales. Each has its advantages, but the Duodecimal Scale is preferable, in this respect, that the transition to it, from the Decimal Scale, would be more practicable, because it would be attended with less violence and difficulty. At any rate, if no change should ever take place, it is a fortunate circumstance, that the Decimal Scale is the most valuable, after the Duodecimal and Trigesimal Scales; and that the present system of Arithmetic possesses so nearly the most perfect kind of notation that numbers can admit.

POSITION.

POSITION is a rule by which the true answer to a question is discovered by means of supposed numbers.

When the answer is found by one supposed number, it is called Single Position; and when two supposed numbers are employed, it is called Double Position.

SINGLE POSITION.

RULE.

TAKE any number, and try if it answer the conditions of the question; if it do, it is the answer; if not, say, as the result derived from this number is to the true result, stated in the question, so is the number supposed to the answer*.

EXAMPLE.

What number is that, which being increased by,, and of itself, the sum will be 155?

Suppose 24
= 12

= 8

18

62: 155: 24: 60. Answer.

EXERCISES.

2. A person, after spending and of his money, had £60 left, what had he at first?

* This rule is founded on the principle, that the results are proportional to the suppositions; which is so obvious, as to require no demonstration.

3. What sum, lent out at 5 per cent., will amount to £350, in 8 years?

4. The joint stock of 2 partners, A and B, was £1250, of which B advanced £160 less than A; required the stock of each?

5. Divide 1142 guineas among A, G, and D, in such a manner, that G may have 106 more than A, and D 58 more than A and G together?

6. A gentleman bought a chaise, horse, and harness, for £60; the price of the horse was double the harness, and the chaise double that of the horse and harness, what was the price of each?

7. Divide 436 guineas among A, B, and C, in such a manner, that B may have 18 more than of A's share, and C 16 more than of B's.

DOUBLE POSITION.

RULE.

ASSUME any two numbers, and work with each of them, as directed in Single Position, and mark the error of each result with the sign+, if in excess, and with —, if an error of defect, then multiply the first position by the second error, and the second position by the first error, and divide the difference of their products by the difference of the errors, when the signs are alike; but the sum of the products by the sum of the errors, when the signs are unlike, which will quote the answer.

EXAMPLE.

A farmer kept a servant for every 60 acres he possessed, and