11. What is the tenth called; and what is its value and use! 12. In what proportion, or ratio, do whole numbers increase; and in what order? 13. What is the effect of this tenfold increase from right to left? 14. What gave rise to the terms adopted in Numeration; and why are not any other terms equally appropriate? 15. What benefits result from this decuple, or tenfold proportion? 16. What other considerations harmonize with the denominations of tens? 17. Whence were modern numbers derived; and why numbered and written from right to left? 18. How could any considerable quantity be expressed, without a tenfold ratio? 19. Of how many values are figures susceptible; and what causes these important distinctions? 20. Why, and how is the local value of a figure justly estimated? 21. What are the terms in Notation, taken in their natural order? 22. How many figures are used under the respective terms of millions, billions, &c.; and what terms are severally prefixed, in regular gradation, to these respective terms; and why? 23. What numbers constitute billions, trillions, &c.? 24. How may any sum be accurately expressed by Notation and Numeration, without liability to mistake? 25. What difference, of actual value, is there between two, or any increased number of significant figures, occupying an equal number of places in Notation? 26. What benefit is derived from a strict observance of the value of which the significant numbers are expressive? 27. What figure, in a given sum, is chiefly expressive of the real value? 28. Whence is the term Decimal derived; or on what principle are Decimals regulated? 29. How do they differ from whole numbers; and by what mark are they distinguished? 30. Into what parts would an unit, or whole number, be di vided, of which the Decimal would be the expression of a part. and what would be the appropriate name given to these divi sional parts of unity? 31. Is it necessary to express the denominator, to obtain the value of the decimal expression? 32. How is the value of decimal numbers obtained? 33. How are Decimals numerated, and why are the terms used? 34. What is the first, or left-hand figure called; and why? 35. Would the name given to the right-hand figure of the Decimal, in Numeration, express the number of parts of which the denominator would consist, if the denominator were expressed? 36. Which figure of a Decimal expresses its principal value; and how many figures will ordinarily express the value with sufficient accuracy? 37. To what extent would Decimals approximate towards each other without meeting; or when would 5 tenths, by annexing nines to it, be made to equal 6 tenths? 38. What effect is produced to Decimals by annexing ciphers; and why? 39. What is the effect of prefixing ciphers to Decimals; and why? 40. How can whole numbers and Decimals be combined together, and yet accurate answers be obtained? INCREASE OF NUMBERS. In all the operations of Arithmetic, numbers are either increased, or decreased. The INCREASE of numbers comprises the rules of ADDITION and MULTIPLICATION; the latter of which is only a compendious method of executing the former. As they are equally concerned in the increase of quantities, it is expedient to treat of them together, by means of which their similarity is more readily seen, and their properties more easily understood by the learner. Their appropriate names will be still retained as before, together with the respective terms peculiar to each, which longestablished usages have sanctioned. When taken collectively, they fall under the general name of Increase; but referred to individually, they are called by their appropriate names of ADDITION Simple Addition teaches to collect several numbers of the same denomination into one quantity. The result of this operation is invariably called sum total, or amount. These terms are never applicable to any other rule, but belong exclusively to Addition, and should be retained in mind. RULE. 1. Place the figures of the several given sums respectively under each other, carefully observing to have units stand under units, tens under tens, &c. and draw a line underneath. 2. Commence with the righthand column, and add the units belonging to it together. 3. Set down under the unit's place the amount, if less than ten. 4. If the amount exceed ten, then set down the right-hand figure of the amount, under unit's place, and add the lefthand figure, or figures to the next column, or row of tens; and thus proceed to the left AND MULTIPLICATION. Simple Multiplication teaches to increase, or repeat the greater of two given numbers as often as there are units, or ones in the less number. It hence performs the work of many additions in a summary manner. The terms belonging to Multiplication are the following: Multiplicand, the number to be increased, or multiplied. Multiplier, the number by which the increase is produced, or the multiplying number. duced by involving or multiProduct, is the number proother; or the result of the mulplying one number into antiplication. The Multiplicand and Multiplier are frequently called Factors, or Terms. RULE. When the multiplier is not over 12, place the multiplier under the multiplicand, so that units stand under units, tens under tens, &c. Multiply each figure in the multiplicand by the whole nultiplier; place the right-hand figure of the product under hand column, where the whole amount is set down. Proof. Add the several columns downwards in the same order they were added upwards; if right, the last sum total will equal the first. Or cut off the upper line, and find the amount of the residue; after which, add the upper line to the last amount, and if this equal the first amount, the work is supposed to be right. Note. The two proofs already suggested, may not at all times prove the result to be correct. The first, of adding the figures downward, is designed rather as a revision of the first addition, by taking the figures in an inverted order, and thereby to prevent the miscalling, or amount of numbers. The same is applicable also to the second rule. Its design is to take the given sums in different orders, so as by revisions, to prevent mistakes. Proof by the Rejection of Nines. RULE. 1. Reject the nines in each of the given sums, and place the excesses severally at the right-hand of the given sums, forming one column. 2. Add these several excesses together, reject the nines, and place the excess below. 3. Reject the nines in the sum total, and if this excess equal the last excess, viz. that of remainders of excesses, the work is supposed to be right. unit's place, and add the lefthand figure, or figures, to the next product arising from multiplying the second figure of the multiplicand. Thus proceed through the multiplicand, and set down all the last product. PROOF. Multiply the multiplier by the multiplicand. Note. The absurd method of proof by Division, when the learner possibly had never heard of the rule, much less of its principles, is now become obsolete. Yet when Division is understood, it affords a substantial rule for the proof of Multiplication. Proof by rejecting Nines. 1. Reject the nines from both factors, and place the excesses severally at the right hand of each. 2. Multiply these excesses together, and reject the nines from their product. 3. Reject the nines from the total product, and if this excess equal the last excess, the work is supposed to be right. Note. The similarity of the nature of the rules of Increase, is obvious, from the preceding examples. It is observable, that when a column of units, tens, or any other, is added or multiplied together, the right-hand figure of the sum total, or product, is placed directly under the same column, or place: and the left-hand figure, or figures, are carried to the next column, or to the product of the next multiplication as so many tens. Suppose the amount, or product, to be 125: the right-hand figure 5, is placed under the column added, or multiplied; and the two left-hand figures, viz. 12, are added to the next column, or to the product of the next multiplication, as so many tens. Such a method of carrying by tens, is far more easy and rational than that of compelling the learner to anticipate the rule of Division, of which he has no knowledge whatever, and thus to ascertain the number of tens, in the amount, or product. It is hence evident also, that whole numbers bear not only a tenfold ratio to each other, but that it is likewise immaterial whether the tens are obtained by addition or multiplication. If we add |