Increment

“ If you once commit the addition table thoroughly, you have it fixed for life; whereas if you do not, you have to make the calculation over again every time, and thus lose a vast amount of labor. I have no doubt that there are some in this class who are in the habit of counting, who have ascertained that seven and eight, for instance, make fifteen, by counting up from seven to fifteen hundreds of times. ”

“ As there is 2⁄5 in the denominator, we must take the part of which 2⁄5 will be a unit. That part is larger than a unit; it is 21⁄2 units; 21⁄2 is that of which 2⁄5 is 1. The above fraction then directs us to repeat 21⁄2 units 31⁄4 times. By extending our word ‘multiplication’ to the taking of a part of a time, all multiplications are also divisions, and all divisions multiplications, and all the terms connected with either are subject to be applied to the results of the other.
If 21⁄3 yards cost 31⁄2 shillings, how much does one yard cost? ”

“ Forming, then, a series of fractions whose denominators are the products of the number of persons indicated in the table of mortality as living at the age of that one who settles the annuity, by the successive powers of unity augmented by the rate of interest, and whose numerators are the products of the annuity by the number of persons living at the same age augmented successively by one year, by two years, etc., the sum of these fractions will be the amount required for the life annuity at that age. ”

“ If we consider that we are dealing with numerical quantities, we must recognize the fact that, while addition and subtraction might in the first instance be limited to such quantities, multiplication and division necessarily introduce the idea of pure number. (ii) If on the other hand we regard ourselves as dealing with pure number throughout, then, as multiplication is continued addition, we ought to include in our classification involution as continued multiplication. ”

“ If one reduces the fraction whose numerator is unity and whose denominator is unity plus a variable, in a series prescribed by the ratio to the powers of this variable, it is easy to see that in supposing the variable equal to unity the fraction becomes 1⁄2, and the series becomes plus one, minus one, plus one, minus one, etc. ”

“ Whatever has quantity, or is measurable, must be made up of parts, which bear proportion to one another and to the whole; so that it may be increased by addition of like parts, and diminished by subtraction, may be multiplied and divided, and, in a word, may bear any proportion to another quantity of the same kind, that one line or number can bear to another. That this is essential to all mathematical quantity, is evident from the first elements of algebra, which treats of quantity in general, or of those relations and properties which are common to all kinds of quantity. ”

“ Counting is fundamental, and counting that is spontaneous, free from sensible observation, and from the strain of reason. A study of these original methods shows that multiplication was developed out of counting, and not from addition as nearly all textbooks treat it. Multiplication is counting. When children count by 4's, etc., they accent the same as counting gymnastics or music. When a child now counts on its fingers it simply reproduces a stage in the growth of the civilization of all nations. ”

“ If we divide 9 into 9 equal parts, each is 1, and 7 of the parts are 7; consequently the fraction which 7 is of 9 is 7/9. The fourth view follows immediately: For a time is only a word used to express one of the repetitions which take place in multiplication, and we allow ourselves, by an easy extension of language, to speak of a portion of a number as being that number taken a part of a time. ”

“ I shall now teach you how to multiply the unknown numbers, that is to say, the roots, one by the other, if they stand alone, or if numbers are added to them, or if numbers are subtracted from them, or if they are subtracted from numbers; also how to add them one to the other, or how to subtract one from the other.
Whenever one number is to be multiplied by another, the one must be repeated as many times as the other contains units. ”

“ The place occupied, in the intellectual development of man, by finger counting and by the many other artificial methods of reckoning,—pebbles, shells, knots, the abacus, etc.,—seems to be this: The abstract processes of addition, subtraction, multiplication, division, and even counting itself, present to the mind a certain degree of difficulty. To assist in overcoming that difficulty, these artificial aids are called in; and, among savages of a low degree of development, like the Australians, they make counting possible. ”

“ With respect to these different rates of increase we may remark, that the first is the most stable of all, and that though its exact quantity is difficult to be ascertained, yet, whatever it is, it is nearly invariable, and, if it can be rightly assumed to give a rate of duplication in ten years at any particular time and place, the same assumption will be equally applicable to all times and places. ”

“ SAPIENTIA. When numbers of any magnitude are set down in order. the first set down is called the "minor term" and the last the "major." When, in making a division, we say by how many the number is to be divided, we give the "divisor," but when we enumerate how many there are in each of the parts we set forth the"quotient." ”

“ The calculation of a logarithm can be performed by successive divisions; evolution requires special methods.
The above definitions of logarithms, &c., relate to cases in which n and p are whole numbers, and are generalized later.
44. Law of Indices.—If we multiply np by nq, we multiply the product of p n’s by the product of q n’s, and the result is therefore np+q. Similarly, if we divide np by nq, where q is less than p, the result is np−q. Thus multiplication and division in the power-series correspond to addition and subtraction in the index-series, and vice versa. ”

“ To do this, consider what the ordinary processes of multiplication and division mean in reference to concrete objects. If we want to give, to 5 boys, 4 apples each, we are said to multiply 4 apples by 5. We cannot multiply 4 apples by 5 boys, for then we should get 20 “boy-apples,” an expression which has no meaning. Or, again, to distribute 20 apples amongst 5 boys, we are not regarded as dividing 20 apples by 5 boys, but as dividing 20 apples by the number 5. The multiplication or division here involves the omission of the unit “boy,” and the operation is incomplete. ”

“ Thus, a fraction is denoted by two numbers parted by a horizontal line; the lower, called the denominator, denotes the number of equal parts into which the integer is divided; and the upper, called the numerator, denotes the number of these equal parts which are taken. Hence it follows, that if the numerator be less than the denominator, the fraction is less than unity. If the numerator be equal to the denominator, the fraction is equal to unity; and if greater than the denominator, it is greater than unity. ”

“ The numerator, in life, is What We Have. The denominator is What We Think We Ought to Have. Mankind may be divided into two classes: Fools and Wise. The fools are eternally trying to get happiness by multiplying the numerator, the wise divide the denominator. They both come to the same—only one you can do and the other is impossible.
If you have only one thousand dollars and think you ought to have two thousand dollars, the answer is one thousand divided by two thousand, which is one half. ”

“ Once more, the child is made to count, and in the process his conceptions of number are cast into a decimal shape, one in which each higher factor is made up of ten of the next lower, till he comes to feel that such tenfoldness is a natural characteristic of enumeration. ”

“ A measure, then, is a divisor which leaves no remainder.
92. When one number is a measure of two others, it is called a common measure of the two. Thus, 15 is a common measure of 180 and 75. Two numbers may have several common measures. For example, 360 and 168 have the common measures 2, 3, 4, 6, 24, and several others. Now, this question maybe asked: Of all the common measures of 360 and 168, which is the greatest? The answer to this question is derived from a rule of arithmetic, called the rule for finding the greatest common measure, which we proceed to consider. ”

“ How can number 45 be divided into four such parts that, if to the first part you add 2, from the second part you subtract 2, the third part you multiply by 2, and the fourth part you divide by 2, the sum of the addition, the remainder of the subtraction, the product of the multiplication, and the quotient of the division, be all equal? ”

“ Numerical quantities, to be added or subtracted, must be in the same denomination; we cannot, for instance, add 55 shillings and 100 pence, any more than we can add 3 yards and 2 metres.
31. Relative Position in the Series.—The above method of dealing with addition and subtraction is synthetic, and is appropriate to the grouping method of dealing with number. We commence with processes, and see what they lead to; and thus get an idea of sums and differences. If we adopted the counting method, we should proceed in a different way, our method being analytic. ”

“ Thus, in seeking our definition of number we have arrived so far at the result that numbers are properties of general terms or general descriptions, not of physical things or of mental occurrences.
Instead of speaking of a general term, such as “man,” as the subject of which a number can be asserted, we may, without making any serious change, take the subject as the class or collection of objects—i.e. “mankind” in the above instance—to which the general term in question is applicable. ”

“ When we employ the multiplication-table in multiplying numbers of several places, and so use the results of old operations of counting instead of performing the whole of each operation anew; when we consult our table of logarithms, replacing and saving thus new calculations by old ones already performed; when we employ determinants instead of always beginning afresh the solution of a system of equations ”

“ A short rule may be found for adding together any number of terms of a continued proportion. Let it be first required to add together the terms 1, r, rr, &c. where r is greater than unity. It is evident that we do not alter any expression by adding or subtracting any numbers, provided we afterwards subtract or add the same. ”

“ Our present method was known specifically as Italian division, and was not introduced until the beginning of the last century.
One writer on arithmetic, a pious monk, furnishes a good illustration of mediæval logic. He is embarrassed by the usage and meaning of the term "multiplication" in the case of fractions in which the product is less than the multiplicand, and he proposes the question, "Whether the multiplication of fractions is an increasing process?" ”

“ Hence we know that in all cases we are justified in neglecting the small quantities of the second—or third (or higher) —orders, if only we take the small quantity of the first order small enough in itself.
But, it must be remembered, that small quantities if they occur in our expressions as factors multiplied by some other factor, may become important if the other factor is itself large. Even a farthing becomes important if only it is multiplied by a few hundred. ”

“ If we are engaged in teaching addition combinations, we should be absolutely certain that we have taught every possible combination. If we want to be sure that children know how to write numbers up to one million, we must have given them drill on all of the possible difficulties. If children are always to respond correctly when problems involving two steps in reasoning are presented, we must have been careful to provide for the purpose of drill all of the combinations of situations involving addition, multiplication, subtraction, and division which can occur. ”

“ The principal multiplied by its time and divided by the interest, and the quotient being multiplied by the factor less one is the time. The sum of principal and interest divided by unity added to the interest on unity is the principal. ”

“ Direct and inverse ratios have been applied by an ingenious author to measure human affections, and the moral worth of actions. An eminent Mathematician attempted to ascertain by calculation, the ratio in which the evidence of facts must decrease in the course of time, and fixed the period when the evidence of the facts on which Christianity is founded shall become evanescent, and when in consequence no faith shall be found on the earth.—Reid, Thomas. ”

“ Quantities, and the ratio's of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer the one to the other than by any given difference, become ultimately equal. ”

“ Upon what Principles, and from what natural Causes do Mankind multiply? And what are the effects of Procreation in Countries where Numbers are not found to increase? The multiplication of mankind has been treated of in different ways; some have made out tables to shew the progression of multiplications, others have treated the question historically. The state of numbers in different ages of the world, or in different countries at different times, has been made the object of inquiry ”

“ Few tests of relative stages of civilisation are more trustworthy than that of the definite conception of high numbers. The prevalence of a decimal system of numerals among widely severed nations, alike in ancient and modern times, has been universally ascribed to the simple process of counting with the aid of the fingers. ”