Thus, 6 is a perfect number, for its aliquot parts are 1(=-=-= 1 3 of 6) 2 (=—=— of 6) and 3 (=- 2 13. An imperfect number is that which is greater or less than the sum of its aliquot parts; in the former case it is called an abundant number, in the latter, a defective number. Thus, 8 and 12 are imperfect numbers; the former (viz. 8) is an abundant number, its aliquot parts being 1, 2 and 4, the sum of which 1+2+4=7, is less than the given number 8. The latter (viz. 12) is a defective number, its aliquot parts being 1, 2, 3, 4, and 6, the sum of which, viz. 16, is greater than the given number 12. 14. A pronic number is that which is equal to the sum of a square number and its root Thus, 6, 12, 20, 30, &c. are pronic numbers; for 6=(4+ √4) 4+2; 12=(9+√/9=) 9+3; 20(16+√16=) 16 +4; 30 (25+ √25) 25+5, &c. Property 1. The sum, difference, or product of any two whole numbers, is a whole number. This evidently follows from the nature of whole numbers, for it is plain that fractions cannot enter in either case. COR. Hence the product of any two proper fractions is a fraction. 2. The sum of any number of even numbers is an even number. Thus, let 2 a, 2 b, 2 c, &c. be even numbers. (See def. 7. cor.) Then 2a+2b+2c+, &c.=their sum; but this sum is evidently divisible by 2, it is therefore an even number; def. 6. COR. Hence if an even number be multiplied by any number whatever, the product will be even. 3. The sum of any even number of odd numbers is an even number. Thus, (def. 7. cor.) let 2a+1, 2b+1, 2c+1, and 2 d+1, be an even number of odd numbers. Then will their sum 2 a+2 b+2c+2d+1+1+1+1, be an even number; for the former part 2a+2b+2c+2d is even, by def. 6. and the latter consisting of an even number of units is likewise even; wherefore the sum of both will be even, by property 2. COR. Hence if an odd number be added to an even, the sum will be odd. 4. The sum of any odd number of odd numbers, is an odd number. For let 2a+1, 2 b+1, 2c+1, be an odd number of odd numbers, then 2a+2b+2c+1+1+1=their sum, the former part of which 2 a+2 b+2c, being divisible by 2, (def. 6.) is an even number, and the latter part 1+1+1, consisting of an odd number of units, is odd: now the sum of both, being that of an even number added to an odd, will, by the preceding corollary, be an odd number. 5. The difference of two even numbers, will be an even number. For let 2 a and 2 b be two even numbers, then since 2a-2 b and 2 b+2a will each be divisible by 2, it is plain that the difference of 2a and 2b will be even, whichever of them be the greater. 6. The difference of two odd numbers is even. For let 2a+1 and 2 b+1 be two odd numbers, whereof the former is the greater; then since 2a+1-2b+1=2a-2b is the proposed difference, which is divisible by 2, it is therefore an even number. 7. The difference of an even number and an odd one will be odd, whichever be the greater. Let 2 a be an even number, 2b+1 an odd number greater than 2 a, and 2c+1 an odd number less than 2 a; wherefore (2 b +1−2 a=) 2 b−2a+1=the difference, supposing the odd number to be the greater; and (2—2c+1=) 2 a−2 c-1=the diffe`rence, supposing the even number the greater. Now each of these differences differs from the even numbers 2 b−2 a, or 2 a-2 c by unity; the difference therefore in both cases is an odd number. 8. The product of two odd numbers is an odd number. For let 2a+1 and 2b+1 be any two odd numbers, then will (2a+1.2b+1=) 4ab+2b+2a+1=their product; but the sum of the three first terms is evidently even, being divisible by 2, and the whole product exceeds this sum by unity, the product is therefore an odd number. (def. 7.) 9. If an odd number measure an odd number, the quotient will be odd. thus, For let a +1 be measured by b+1, and let the quotient be q ; a+1 b+1 VOL. II. =q; then will b=1.q=a+1; and since b=1, and H a+1 are odd, it is plain that q must be odd, otherwise an odd number multiplied by an even number, would produce an odd number, which is impossible. (proper. 2. cor.) 10. If an odd number measure an even number, the quotient will be eveno. For let 2 a 2b+1 q, then 2b+1.q=2a; and since 2b+1 is Mr. Bonnycastle, in treating on this subject, (Scholar's Guide, 5th Edit. p. 203.) has committed a trifling oversight. Prop. 10. in his book is as follows; "If an odd or even number measures an even one, the quotient will be even." The former position is here shewn to be true, but the latter is evidently false, namely, "if an even number measure an even number, the quotient is even." In proof of his assertion he says, "let 2 a 26 = then 2 b.q=2a; and since 2 and 2 b are even numbers, q must likewise be an even number." This conse→ quence however does not necessarily follow; q may be either even or odd, for any even number (26) multiplying any odd number (7), will evidently produce an even number. (See proper. 3.) Hence the quotient of an even num ber by an even number, may be either even or odd; thus, ber; but 6 2 8 -=3 an odd number. Mr. Keith has fallen into the same error, or (which is more probable) has copied it from the above work. See his Complete Practical Arithmetician, 3d Edition, p. 283. Cor. to Art. 32. The first named Author is likewise mistaken when he says, (Prop. 11.) " If an odd or an even number measures an even one, it will also measure the half of it." Now the half of any number will evidently measure the whole, and the half measures itself, that is, it is contained once in itself; wherefore it follows, according to the tenor of the reasoning there employed, that if one quantity be contained once in another, the former quantity measures the latter, but the whole is contained once in the whole, and therefore measures it: but whatever measures the whole measures its half, says Mr. B. wherefore the whole must necessarily measure the half! This mistake seems to have arisen from a circumstance which might easily have happened-that of confounding the idea of a measure with that of an aliquot part: had it been said that every aliquot part of the whole measures the half, the assertion would have been perfectly accurate. Should the freedom of the above remarks require an apology, I feel it necessary to testify my unreserved admiration of the eminent talents of the learned and respectable authors in question, and to assure them that nothing invidious can possibly be intended: but truth is the grand object of the sciences, and he who is engaged in the arduous and important office of instruction, forfeits all claim to fidelity and confidence, if he does not point out error wherever he may happen to find it; and he is scarcely less blameable who omits to do it with becoming candour, and under a sense of his own fallibility. an odd number, and 2a an even one, it follows that q must be even; otherwise the product of two odd numbers would be even, which is impossible. (proper. 8.) 11. An even number cannot measure an odd number. If possible, let 2a+1 -=q; wherefore 2a+1=2b.g: but since 2b is an even number, 2b.q is also even, (proper. 2. cor.) that is, an odd number (2a+1) is equal to an even one, (2 b.q,) which is absurd: wherefore an even number, &c. 12. If one number measure another, it will measure every multiple of the latter. Let n any whole number, and 응=9, na then will =nq. But since q is by hypothesis a whole number, nq must be a whole number, (proper. 1.) that is, b measures n times a. 13. That number which measures the whole, and also a part of another number, will likewise measure the remainder. 14. If one number measure two other numbers, it will likewise measure their sum and difference. Let c measure both a and b, then will and be both a b C C COR. Hence the common measure of two numbers will like wise be a common measure of the sum and difference of any multiple of the one, and the other. 15. If the greater of two numbers be divided by the less, and if the divisor be divided by the remainder, and the last divisor by the last remainder continually, until nothing remain, the last divisor of all will be the greatest common measure of the two given numbers. Let a and b be two numbers, and let a be contained in b, p times with c remainder; let c be contained a) b (p For since (bap+c, or) b-ap=c, and a-qc=d, it follows (from proper. 12.) that every quantity which measures a and b, will likewise measure ap, and also b-ap or c, (proper. 13.) in like manner, whatever quantity measures a and c will also measure a and qc, and likewise (a-qc, or) d; wherefore any quantity which measures d, must likewise measure c and a and b, but d measures d, therefore it is a common measure of a and b. It likewise appears, that d is the greatest common measure of a and b; for since rd=c and (cq+d=) rdq+d=a, and (ap+c=) rdqp+ dp+rd=b, that is, rq+1.d=a, and rqp+p+r.d=b, it follows that d is the greatest common measure of these two values of a and b, or that it is a multiple of all the common measures, except the greatest, of a and b. Otherwise, since it appears that every common measure of a and b measures d, and d itself measures a and b, it follows that d is the greatest common measure of a and b3. 16. The sum and the difference of two numbers will each measure the difference of the squares of those numbers. For since a+b.a-ba-b", it follows that 17. The sum of any two numbers measures the sum of their cubes, and the difference of any two numbers measures the difference of their cubes. P See Wood's Algebra, third Edition, p. 45. The above is a demonstration of the rule in page 148. of the first volume. |