mark all the multiples of 3, 5, 7, etc., with a dot above them, and all the numbers not marked will plainly be prime numbers. Thus we have found 1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101 all to be prime numbers; and by carrying the series of odd numbers sufficiently far, we may find as many prime numbers as we please. (25.). There is a method of determining whether a given number is prime or not, which it may be well to notice. If a number is not prime, it will have two factors at least, which will equal each other when the number is a square; but when the number is not a square, one of the factors will clearly be less than the square root of the number, and the other will be greater. Hence, if we find that the number is not divisible by any prime number that is less than its square root, we may infer with certainty that the number is prime. Thus, to determine whether 79 is a prime number or not, we find that it is not divisible by either of the numbers 2, 3, 5, 7, .. it is a prime number; also 131 is not divisible by either of the numbers 2, 3, 5, 7, 11, .. it is a prime number; but 91 is divisible by one of the numbers 2, 3, 5, 7, viz., by 7, ... 91 is a composite number. We shall soon give another method of finding the greatest common measure, or least common multiple, of two or more numbers, which does not require the use of prime numbers. (26.) Again, in order that an integral quantity may be resolved into any number, n, of equal factors, it is evidently necessary (from what has been done) that it should have been formed by taking the product of n equal factors; if it has not been formed thus, or if it does not consist of the product of n equal factors, it of course can not be resolved into n equal factors, except approximately; and the quantity is called a surd or an irrational quantity. Thus, if we have to take the cube root of the integral quantity a2, supposing a not to consist of three equal factors, it evidently can not be done except approximately, for the cube root requires a2 to be resolved into three equal factors, and one of the factors to be taken; and it is evident that if a2 consists of three equal factors, each of them must equal the square of a factor of which a consists of three that are equal to each other. Consequently, if a is not a cube, or if it does not consist of three equal factors, we say that a2 is an irrational quantity with respect to the cube root; that is, that Va (a), which denotes the cube root is an irrational quantity. Now the square root of a2 is evidently a, ora; so that a quantity which is irrational with respect to a root of one kind, may sometimes be rational with respect to a root of another kind. (27.) To make what has been said more evident, we observe if a denotes any positive integral quantity which does not consist of n equal integral factors, that it does not consist of the product of n equal fractions, whose numerators and denominators are composed of a finite number of figures. I m For, if possible, let an = where m and p are finite and p' integral; by taking the nth power, we get (a)" = a= m Mn pr an impossible result, for the fraction may be supposed to be Р in its lowest terms, or such that its numerator and denominator have no common factor except unity; consequently, from what has been proved, m" and p have no common factor except unity; so that we have the integral quantity a mn equal to the irreducible fraction which is impossible. pr Hence, by (4) of the definitions, vaa", which denotes the nth root of a, is an irrational quantity. (28.) Now, although a can not be expressed by a fraction whose numerator and denominator are finite integers, yet two fractions, whose denominators are equal, and whose numerators differ only by a unit, may be found, so that a" shall be greater than the smaller fraction, and smaller than the greater fraction; and the fractions thus found are said to be limits of a". I For, if we supposer to be the greatest integral n' root contained in ap", when p is supposed to be a positive integer> 1, then, since ap" is not an exact n" power, we shall have ap" and ap" <(r+ 1)", or, which is the same r + 1 are said to be limits of a". If r and p are indefinitely great numbers, it is evident that the fractions rr + 1 p' p will differ infinitely little from each other; for their difference, 1 will become an infinitely small quantity, since p, the divisor of 1 the numerator, is indefinitely great. It is customary in practice to find r and p to a great num I ber of figures, and to take for the value of a", and to say I Р that the value of a" is correctly found to within the fraction I 1 but it is evident that a more correct value of an will often From what has been done, it follows that a can not be expressed by a fraction whose numerator and denominator are finite integers. To illustrate what has been said, we shall find some of the limits of the square root of 2; or, which is the same, shall find limits for 12. = If we assume p = 10, then 2p2 200, and the greatest integral square contained in 200 is 196 142, .. √2 14142 14143 10000' 10000 are nearer limits, and are yet nearer limits of 2; and in this man ner we may proceed indefinitely to find nearer limits of 1/2 (29.) Again, if we represent any irrational quantity by a, and let m denote any very small rational quantity of the same kind, as a, then by taking m, 2m, 3m, 4m, etc., we may evidently find some whole number, as p, such that the inequalities, pm <a, pm + m > a shall have place; then a will have pm and pm + m for its limits. Now it is manifest that by supposing m to be very small, and consequently p very great, we may bring these limits as near to equality as we please; for we may suppose m to be diminished ad infinitum, and p increased accordingly. In like manner, if b represents any irrational quantity, and n a very small rational quantity of the same kind as b, we may find the integer q such that the inequalities qn <b, qn+nb shall have place; in which we may suppose q to be taken so great that n, the difference of the limits qn, qnn, of b, shall be less than any given quantity. = (30.) We are now prepared to show that the product ab is equal to the product ba; which may be done as follows, viz., since pm is less than a, and qn less than b, we have pm × qn < ab and qn × pm <ba, and since pm, qn are rational quantities, we have, by what has been proved in Multiplication and Division, pm × qn = qn × pm, ... pm × qn < ab and pm x qn <ba have place at the same time. We may show, in a similar way, that the inequalities (pm + m). (qn + n) > ab, (pm + m). (qn + n) > ba have place at the same time. Hence, the products ab, ba have the same limits, which are pm × qn and (pm + m). (qn+n) (pm + m). qn + (pm + m) . n = pm . qn + pm . n + qn. m +m.n; in which pm, qn represent rational finite quantities, and m, n indefinitely small quantities. Now pm.n+ qn.m + m. n is the difference of the limits, which will be diminished indefinitely by diminishing m and n indefinitely; for since the finite quantity pm is multiplied by the indefinitely small quantity n, the product pm.n will be indefinitely small; in the same manner the product qn.m is indefinitely small, and the product m.n is also indefinitely smaller than the other products, since each of its factors is indefinitely small. Hence, evidently, by supposing m and n to be less than any given quantities, the difference of the limits will be less than any given quantity'; consequently, since ab and ba differ less from each other than their limits, it follows that they differ less from each other than any given difference; consequently (Ax. VII.) we must have abba, as required. A similar demonstration is applicable when one of the quantities, a, b, is rational, and the other irrational. And in like manner it may be shown that the product of any number of irrational quantities, or of quantities some of which are rational and others irrational, is independent of the order of the factors. (31.) Since all real quantities are either rational or irrational, it follows, from what has been shown, together with what was formerly proved, that the product of any number of real quantities is independent of the order of the factors. Hence, according to custom, the product of literal quantities may be indicated by writing the letters alphabetically. Thus, the product of a, v, s, t, x is expressed by astvx, and bpqr expresses the product of r, b, q, p. (32.) OF THE GREATEST COMMON MEASURE OR DIVISOR. The product of all the particular divisors that are common to two or more numbers or quantities of the same kind, is called their greatest common measure or divisor. Thus, if we have two numbers or quantities, am and bm, such that m is their greatest common divisor, then a and b can have no other divisor than 1; for if they can, m will not be the greatest common divisor of am and bm, which is against the hypothesis. If we suppose am to be greater than bm, and divide am by bm, using p to represent the integral part of the quotient, then we shall have the remainder of the division expressed by am — pbm pbm = (a - pb)m, which, by the nature of division, is less than the divisor bm. Since a and b can have no other common divisor than 1, it is clear that b and a - pb can have no other divisor than 1; for if they can, since it divides b and a-pb, and as it divides b in pb, it must divide a, for otherwise it can not divide a pb; hence, if b and a-pb can have any other common divisor than 1, a and b must also have a common divisor other than 1, which is against the hypothesis. It hence appears that m is the greatest common divisor of bm and the remainder of the division of am by bm, which is |