BOOK II. THEORY OF ARCHITECTURE. CHAP. I. A RITH METIC AND ALGEBRA. SEct. I. introduction. 522. The abstract relations of quantity and figure ought to be thoroughly understood by the architect, that he may be able to prepare the designs which he has conceived, in a manner suitable for execution, and, when executed, to possess stability. The form and mechanical effect of each single block in a building depends on its position, and the form of one of its parts limits the forms of others. In groups of bodies, these limitations are still more perplexing; hence we must have recourse to the most easy and accurate means of ascertaining the practicable conditions which will produce the desired effect. To this end we propose to give a short and simple course of the elements of arithmetic and analysis, as our own experience informs us that occasions arise in the practice of architecture which require all the aid that science can afford. Those who have studied most closely, know that they have not acquired too much ; whilst those who have not studied at all have to depend on the skill of others, and, like all similar dependents, become more or less the dupes of those they employ. 523. That which is capable of increase or diminution is called magnitude or quantity. Hence the different kinds of magnitude must be many. Mathematics, generally speaking, is the science of quantity, or that which investigates the means of measuring quantity. Now we cannot measure or determine any quantity except by pointing out its relation to some other known quantity, so that the determination or the measure of magnitudes of all kinds is the making any one known magnitude of the same species the measure or unit for determining the proportion of the proposed magnitude to this known measure. This proportion being always expressed by numbers, a number is but the proportion of one magnitude to another, arbitrarily assumed as the unit. Hence all magnitudes may be expressed by numbers, and the foundation of all mathematical science must be laid in a study of the science of numbers, and in an examination of the different methods of calculation involved in it. In Algebra, or analysis numbers, which represent quantities, are alone considered, without respect to the different kinds of quantity. The latter are the subject of other branches of mathematics. Arithmetic is the science of numbers properly so called, extending only to certain methods of calculation which occur in common practice. Algebra comprehends all the cases that can exist in the calculation of numbers. signs + AND -. 524. (1.) When one number is to be added to another, the sign + (plus) is used, and is placed before the second number. Thus, 5 + 3 denotes that 3 is to be added to the number 5, the sum whereof every body knows to be 8. The same sign may be used to connect several numbers, thus 7 # 9 + 12 + 81 signifies that to the number 7 we must add 9, 12, and 81, which make 109. All this is evident, but in Algebra, in order to generalise numbers, they are represented by letters as a, b, c, d, &c.; thus a + b + c + d signifies the sum of the numbers represented by those letters. 525. (2.) To subtract one number from another the sign – (minus) is used, which is placed before the number to be subtracted; thus 10–6 signifies that the number 6 is to be taken from the number 10, so that the expression is equivalent to the number 4. So of several numbers; as, for instance, 62–6–15–31 signifies that 6 is to be take from 6% 2 the remainder is 56; 15 taken from that remainder leaves 41 ; lastly, take from this 31, and the remainder is 10, which is the value of the expression. We might, however, have taken the sum of the numbers 6, 15, and 31 or 52 at once from 62, and the remainder is 10, as before. 526. It is easy, therefore, to determine the value of expressions in which both the signs + plus and – minus are found; for example, 16–4–7 + 3–1 is the same as 7. For we have only to collect the numbers with the sign + before them, and subtract from their sum those that have the sign —. Thus, the sum of 16 and 3 is 19 ; that of 4, 7, and l is 12; and 12 being taken from 19 the remainder is 7. It must be remembered that in the expression the sign + is supposed to stand before the number 16 ; and that the above expres. sion might have been arranged thus: 16 + 3–4–7 – 1, or 3 — I – 4 – 7 # 16, or 3 + 16-1 –7–4. If, instead of numbers, we use letters, no more difficulty occurs; for example — signifies that certain numbers are expressed by a and d, and that from them or their sum the numbers expressed by the letters b, c, and e, having the sign – before them, are to be subtracted. Hence attention is necessary to the sign prefixed to each number, for in algebra simple quantities are numbers considered with respect to the signs which affect them. Those quantities before which the sign + is found are called positive quantities, and those affected by the sign —are called negative quantities. To illustrate this, let us suppose a man having iOOOl., but owing 400l., it is evident his real wealth is only 1000l. – 400l., 600l. Thus, negative numbers may be considered as debts, because positive numbers represent real possessions, and we may, indeed, say that negative numbers are less than nothing. For, take a man having nothing, and at the same time owing 100 pounds, it is clear he has 100 pounds less than nothing; for, if he had a present of 100 pounds made him to pay his debts, though he would be richer than before, he would still be at the point nothing. So, therefore, as positive numbers are clearly greater than nothing, negative numbers are less than nothing. Now, positive numbers are obtained by adding 1 to 0, that is, to nothing, and by thus increasing them from unity. This is the origin of the series called natural numbers, of which the following are the leading terms of the series: o + 1 + 2 + 3 + 4 +5+ 6 +7 + 8 + 9 + 10, and so on to infinity. But if, instead of adding, we perpetually subtract unity, we have a series of negative numbers, thus: 0–1–2–3–4 –5–6-7–8–9–10, &c. to infinity. These numbers, whether positive or negative, are called whole numbers or integers, either greater or less than nothing. They are so called to distinguish them from fractions and other kinds of numbers, which will be hereafter noticed. Thus, between 2 greater by a unit than 1, it is easy to conceive an infinity of numbers greater than 1, yet all less than 2; for imagine a line of 2 ft. long and another 1, it is evident that an infinite number of lines may be drawn, all longer than 1 ft., but not so long as 2 ft. That a precise idea may be formed of negative quantities, the reader must keep in mind that all such expressions as +1 –1, + 2–2, + 3–3, &c. are equal to 0. and that +2–5 is equal to -3. For, if a person has 2 pounds and owes 5, he has not only nothing, but still owes 3 pounds; and the same observation holds true with respect to letters, which represent numbers, thus + a -a is O. But, if the value of + a -b is wanted, two cases are to be considered : first, if a is greater than b it must be subtracted from a, and the remainder, with the sign + placed or understood before it, is the value sought; secondly, if a is less than b, a is to be subtracted from b, and the remainder must have the negative sign placed before it. MULTIPLICATION OF SIMPLE QUANTITIES. 527. (3.) In the multiplication of simple quantities, where two or more equal quantities are added together, the expression of their sum may be abridged thus : — where x is the sign of multiplication. thus we obtain an idea of multiplication; for in the above, it must be observed tha' 9 x a signifies 2 times a, 3 - a signifies three times a, 4 x a four times a. But this form is abbreviated by simply putting the number before the letter; thus, a multiplied by * * expressed 3a b multiplied by 5 is 5b : c taken but once, or 1 c, is no more than c. Hence the multiplication of such products by other numbers is simple enough, for 528. Suppose both the numbers be represented by letters, we have only to place one before the other, and the process is complete; thus a multiplied by b is ab; and if again we multiply this product by po, the result is abpg. The order of the letters is no consequence; for suppose a to represent 5, and b 6, then ba or ab equally represent 6 × 5 and 5 x 6, which give the same product. But in the use of common numbers this cannot be done; for were we to write 34 for 3 times 4, we should have 34 instead of 12. If the sign x is omitted, it is usual to place a point between the figures; thus, 1 . 2. 3. 4. 5 is 120, as is 1 × 2 × 3 × 4 x 5. Hence, if we meet with the expression 2. 3.4 ryz, it means that 2 is to be multiplied by 3, and the product by 4; and that product first by r, then by y, and lastly by 2, hence this may be abridged into 24 a yz. 529. The result arising from the multiplication of two numbers is called a product, and the numbers or letters are called factors. 530. In the case of positive numbers being multiplied into each other, no doubt can remain of the products being positive, for + a x + b must necessarily give ab. But the multiplication of + a by -b, or of — a by —b, requires examination. Suppose – a multiplied by 3; now, as – a may be taken as a debt, if multiplied by 3 it is three times greater; hence the product must be –30. And if that be multiplied by + b, it is evident the debt is still increased by the action of b upon it; it becomes —ba, or, which is the same thing, —ab. On this account it is evident that if a positive be multiplied by a negative quantity, the product becomes worse, or, if the expression might be allowed, more negative. From this follows the rule, that + by + is always plus, and that + by —, on the contrary, gives a minus quantity. But the case in which – is multiplied by minus, that of —a by -b, requires consideration. There can be no doubt that the product is ab; the sign, however, to be prefixed to it is at first sight not so clear. Now we have seen that it cannot be —, for — a multiplied by + b gives —ab, and – a by -b, cannot produce the same result as —a by + b ; hence it must produce a contrary result, that is + ab, and hence results the following rule: – multiplied by — produces + just in the same manner as + by +. This is more briefly expressed in the following terms, Like signs multiplied together give +, unlike or contrary signs give —, whereof take as an example the multiplication of the following numbers, + a, -b, -c, + d. First + a multiplied by —b makes —ab, this by c gives +abc, and this last by + d gives + abcd. 531. It remains only to show how to multiply numbers that are themselves products. Now, to multiply the number ab by the number cd, it is manifest, from what has been said, that the product is abcd, and that it is obtained by first multiplying ab by c, and the product by d. Or, if we had to multiply 36 by 12, 12 being equal to 3 times 4, we should first multiply 36 by 3, and the product 108 by 4, in order to have the whole product of the multiplication of 12 by 36, or 432. But, if we have to multiply 5ab by 3ry, we may write 3ry x 5ab; but, as the order of the numbers is indifferent, it is better, and is the custom, to place the common numbers before the letters, and to express the product thus: 5 x 3abry, or 15abry; 5 times 3 being 15, so 6abc by 7+y gives 42abcry. whole NUMBERs in RESPECT to Thiel R FACTORs. 552. A product, as we have seen, is generated by the multiplication of two or more numbers. These are called factors. Thus, abcd are the factors of the product abcd. All whole numbers cannot result from such a multiplication: those which are in that predicament have not any factors. Thus, 4 is produced by 2 x 2, 6 by 2 x 3, 8 by 2 x 2 x 2, 27 by 3 × 3 × 3, &c. But the numbers 2, 3, 5, 7, 11, 13, 17, &c. cannot be represented by factors, unless, for the purpose, we make use of unity, and represent, for instance, 2 by 1 x 2. Now, as numbers which are multiplied by 1 remain the same, unity cannot be reckoned as a factor. Hence, all numbers, such as 2, 3, 5, 7, &c., which cannot be represented by factors, are called simple, or prime numbers, whereas others, as 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, &c., which can be represented by factors, are called compound numbers : simple or prime numbers consequently deserve particular attention, inasmuch as they do not result from the multiplication of two or more numbers; and it is worthy of observation, that in writing these numbers in succession as they follow each other, thus, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, &c., no regular order is perceptible, their increments being sometimes greater, sometimes less, and, as yet, no law which they follow has been discovered. 553. All compound numbers which may be represented by factors have prime numbers for their factors; for if a factor is found which is not a prime number, it may be decomposed and represented by two or more prime numbers. Thus, if we represent the number 30 by 5 x 6, 6, not being a prime number, might have been represented by 2 x 3, that is 5 × 2 × 3, in which the numbers are all prime; numbers equally represent 30. There is much difference between compound numbers, which may be resolved into prime numbers; some have only two factors, others three, and others still more. Thus we have seen that The analysis, therefore, of any number, or the resolution of it into simple factors, is easily accomplished. Take, for instance, the number 360. First, it may be represented by 2 x 180. Then 180 is equal to 2 x 90, and So that the number 360 may be represented by these simple factors, 2 × 2 × 2 × 3 × 3 × 5, since these numbers multiplied together produce 360. This shows that prime numbers cannot be divided by other numbers, and that the simple factors of compound numbers are most conveniently found by seeking the prime numbers, by which compound numbers are divisible. Division of SIMPLE QUANTITIES. 534. The separation of a number into two or more equal parts is called dirision, which enables us to determine the magnitude of one of those parts. For instance, suppose we wish to separate 12 into three equal parts, we find, by division, that each of those parts is equal to 4. The number to be divided is called the diridend, the number of equal parts into which it is to be separated is called the divisor, and the magnitude of one of the parts determined by the division is the quotient : thus, in the example, – 12 is the dividend, From this it is evident that if we divide the number 2 into two equal parts, one of those parts, or the quotient taken twice, is exactly the number proposed; and so, if a number is to be divided by 3, the quotient thrice taken must again give the same number. Hence follows the general rule that the quotient multiplied by the divisor reproduces the diridend. The dividend, indeed, may be considered a product, whereof one factor is the divisor and the other the quotient. For, if we have 40 to divide by 8, we have to find a product in which one of the factors is 8, and another factor which multiplied by it may give 40. Now 5 x 8 is a product which answers the hypothesis, and therefore 5 is the quotient of 40 divided by 8. 535. Generally, a number ab divided by a gives a quotient b, because a multiplied by b gives the dividend ab. So, if we have to divide ab by b, the quotient must be a. In short, the whole operation of division consists in representing the dividend by two factors whereof one is equal to the divisor and the other to the quotient; thus the dividend abc divided by a gives bc, for a multiplied by be produces abc; and, similarly, abc divided by b gives ac, and abc divided by ac gives b. So 16+y divided by 4a gives 4y, inasmuch as 4 times r multiplied by 4 times y produces 16xy; but had 16ay been divided by 16, the quotient must have been ry. 536. A number a is the same as 1a : hence, if a or la is to be divided by 1 the quotient must be the same number a ; but if the same number a or la be divided by a, the quotient must be 1. 537. The dividend cannot always be represented as the product of two factors, whereof one is equal to the divisor; in which case other expressions must be had recourse to. Thus, in dividing 19 by 6, it is obvious that the number 6 is not a factor of 19, for 6 x 3 is but 18, and therefore too small, and 6 x 4 produces 24, which is too large; from which it is evident that the quotient is greater than 3 and less than 4. To determine this exactly, a species of numbers called fractions is used, whereof we shall hereafter treat. But previous to that, let us investigate the number which nearest approaches to the true quotient, with attention to the remainder left, thus : — where the dividend is 19, the divisor 6, the quotient 3, leaving a remainder of 1. Now, if we multiply the divisor 6 by the quotient 3, and thereto add the remainder, we have the dividend, and this proves the correctness of the division ; for 3 multiplied by 6 produces 18, to which, if the remainder 1 be added, we have 19, the dividend. 538. Here it must be observed, in respect of the signs + and —, that +ab divided by + a must be + b ; for it is evident that + a multiplied by + b gives + ab. But, if + ab be divided by —a, the quotient must be —b, because – a multiplied by —b produces + ab. Suppose the dividend — abdivided by + a, the quotient must be —b, because – b multiplied by + a makes —ab. Lastly, the dividend –ab divided by — a must have for its quotient + b, for the dividend — ab is produced by – a by + b. 539. In division, therefore, the same rules hold respecting the signs + and — as in multiplication; namely, + by + give + and + by — give –, — by + give — and — by – give +, 540. Some numbers are, it has been seen, divisible by certain divisors, others are not so. Let us look to this difference between them. Take the divisors 2, 3, 4, 5, 6, 7, 8, 9, 10, &c. 541. Now in the divisor 2 the numbers it will exactly divide are manifestly 2, 4, 6, 8, 10, 12, &c., in which the series increases uniformly by 2, and they are called even numbers. But in the numbers 1, 3, 5, 7, 9, 11, 13, 15, &c. there is an uniformly less or greater number by unity than in the former not divisible by 2 without a remainder 1 : these are called odd numbers. 542. The general expression 2a includes all the even numbers, for they are obtained by successively substituting the integers 1, 2, 3, 4, 5, 6, 7, &c.; and for this reason the odd numbers are comprehended in the expression 2a+ 1, because 2a + 1 is greater by unity than the even number 2a. 543. In the second place, suppose 3 to be the divisor, the numbers it will exactly divide are 3, 6, 9, 12, 15, 18, 21, &c., which numbers are comprehended in the expression 3a, for dividing 3a by 3 the quotient is a without a remainder. All other numbers that we would divide by 3 will give 1 or 2 for a remainder; and hence they are of two kinds: first, those leaving the remainder 1 after the division, which are 1, 4, 7, 10, 13, 16, &c., and are contained in the expression 3a + 1 ; second, those in which 2 is the remainder, and these are 2, 5, 8, 11, 14, 17, 20, and these may be expressed 3a +2; so that all these numbers may be expressed by 3a, 3a + 1, or by 3a +2. 544. Suppose 4 to be the divisor, it will divide the following numbers, 4, 8, 12, 16, 20, 24, &c., which increase uniformly by 4, and are comprehended in the expression 4a. All other numbers not divisible by 4 may leave the remainder 1, or be greater by 1 than the former, as 1, 5, 9, 13, 17, 21, &c., and may be comprehended in the expression 4a + 1 : or they may give 2 as a remainder, as 2, 6, 10, 14, 18, 22, &c., and be expressed by 4a + 2 ; or, lastly, they may give the remainder 3, and as 3, 7, 11, 15, 19, 23, &c., and be represented by the expression 4a +3. All possible integer numbers are hence contained in one or other of the four expressions 4a, 4a + 1, 4a + 2, 4a +3. 545. If the divisor is 5 it is nearly the same, for all numbers divisible by it are comprehended in the expression 5a, and if not divisible by 5 they may be reduced to one of these expressions, 5a + 1, 5a +2, 5a + 3, 5a +4, and so we may go on to the greatest divisors. 546. It is necessary to keep in mind, as we have noticed in a previous passage on the resolution of numbers into their simple factors, that all numbers among whose factors are found 2 or 3, or 4, or 5 or 7, or any other number, are divisible by those numbers. For example, 48 being equal to 2 × 2 × 3 × 4, it is clear that 48 is divisible by 2 and by 3 and by 4. 547. As the general expression abcd is not only divisible by a and b, and c and d, but also by ab, ac, ad, bc, bd, cal; and by abc, abd, acd, bed; and, lastly, by abcd, which is its own value: it is clear that 48, or 2 × 2 × 3 × 4, may be divided not only by those simple numbers, but by those composed of two of them, that is, by 4, 6, 8, 12; and also by those composed of three of them, that is, by 12, 16, 24; and, lastly, by 48 itself. From this it follows, that when a number has been represented by its factor it is easy to find all the numbers by which it is divisible. |