7

Digit of the Minorand from 10, and to the Remainder add the correfpondent Digit of the Subducend; but, in this Cafe, you must remember to add an Unit to the next Digit of the Minorand. Note, It is customary to put the Subducend over the Minorand, but this is entirely arbitrary.

40. Example. From 7540 fubtract 2172.

Subducend 7540

Minorand 2172

Place the Numbers thus:

Remainder 5368

Then, fince you cannot take 2 (the first Figure of the Minorand) from o, (in the firft, or Unit's Place of the Subducend;) fay, 2 from 10, or 10-2=8, which put down; then, adding (you carry) to 7, fay, 8 (from 4 you cannot, and .) from 10, or 10--8=2, +46, which write down; and proceed, faying, 1 (you carry) +1=2, and 2 from 5, or 5-2-3, which put down: Laftly, 2 from 7, or 7-2=5, which, being placed down, gives the Difference or Remainder =5368.

41. When the Tyro has been fome Time exercifed in the above Method, he may learn the following more commodious Way of performing Subtraction, viz. by adding 10 to any Digit of the Subducend, when the correfpondent Digit of the Minorand is greater than that of the Subducend; and then, from. that Sum, fubtracting the correspondent Digit of the Minorand; thus, in the above Example, we may fay, 10-2=8, which fet down; then 1 (we carry) +7=8, and 14-8-6, which write down; then i (we carry) +1=2, and 5-2-3, which place down: Laftly, 7-2=5. Hence the Remainder is 5368 as before.

42. Mr. Lowe thinks Subtraction is better performed by Addition; thus, in the above Example, fay 2 and as much as will make the Amount to the next Row is 8, which 8 put down; then I (we

carry)

9.

47.

carry)+7=8, and 8+6=14, put down 6; and fay, 1 (we carry) +1=2, +3=5, put down 3; then 2+57, put down the 5; which makes the Remainder 5368, as above.

43. A Demonftration of Art. 39. or Rule for finding the Difference of two Numbers; which may be conveniently parted into two Cafes.

1. When all the Figures of the Minorand are lefs than (or equal to) their Correfpondents in the Subducend, it is manifeft, that the Difference of the Figures in the feveral Places, being put in the famePlace as the Figures ftand whofe Differences they are, must, taken all together as one Number, be equal to the Difference fought; for, as all the Parts of any Number taken together are equal to the Whole, fo muft the Differences of all the like Parts of any two Numbers make up the Difference of the Wholes. * 2 E. D.

2. When any Figure of the Minorand is greater than its correfpondent Figure in the Subducend, by the Rule in Art. 39. we fubtract that Figure of the Minorand from 10, and add the Subducend Digit to that Remainder; or, which is the fame, add 10 to the Figure in the Subducend, and take the correfpondent Figure of the Minorand from that Sum; and then add 1 to the next higher Figure in the Minorand. Now, fince by * Notation 10 in any Place is I in the next higher Place, it is evident, we have increased both the Subducend and Minorand with an equal Number; and therefore the Difference of the Subducend and Minorand, fo increased, will be the fame as if they were not increased. 2. E. D.

=

*

The only Thing that remains here to be taken Notice of, is, that, any Digit being taken from the Sum of any leffer Digit and 10, the Difference will never exceed 9; the Reafon of which will plainly appear, by confidering, that as the leffer Digit wants

at

*Note, 2.E. D. amongst Mathematicians, fignifies, Quod erat demonftrandum (which was to be demonftrated); and 2.E.I. is, Quod erat inveniendum (which was to be found),

at least to make it the greater; when we have taken all we can out of the leffer Digit, there muft, at least, remain 1 (of the greater Digit) to be taken from the 10, and therefore the Remainder can never exceed 9.

44. When we are to fubtract one Number" from feveral, or several from one, or several from several, it is beft, before we fubtract, to fhorten the Work, as much as may be, by Addition: Thus, if it be required to take 30+15+16, from 8+23+16+50, the best Method of Operation will be as under:

45. Subtraction may be proved by adding the Minorand and Remainder together; for their Sum must (by Art. 24.) be equal to the Subducend.

46. In Literal or Algebraical Subtraction, all that is neceffary to be observed is, that, Subtraction being directly contrary to Addition, to fubtract an Affirmative must be the fame as to add a Negative; and to fubtract a Negative, the fame as to add an Affirmative: Hence, Subtraction of Algebra may be performed by changing (or fuppofing in your Mind) all the Signs of the Minorand (to be changed); and then adding them as in Addition. Thus, 2x from 5x, remains 5x-2x=3x; and 2x from -5x remains -2x and 2x from 4-5x -5x=-7; alfo

-

+2x+5x=+7x; and 2x from 5x=-3x.

leaves

5x leaves + 2x

47. Before we put an End to this Chapter, it may not be improper here to remark, that, in Art. 43, we took it for granted, that, when equal Quantities be added to both the Subducend and Minorand, the

Difference of thefe Sums will be equal to the Difference of the (former) Subducend and Minorand; which may be thus demonftrated:

Let s the Subducend, m the Minorand, d= their Difference, or s-m-d; a the Number to be added; then we have, for a new Subducend, s+a, and for a new Minorand m+a. Now m from s= s-m, and a from ao; .s+a Minus m+a sm d, per above. QE.D.

=

Note, When a Dash (-) is drawn over any Quantities, it denotes that thofe Quantities are to be taken together; thus, in the above, by s+a Minus m+a, is to be understood, that the Sum of m and a is to be taken from the Sum of s and a.

48. We will put an End to this Chapter, with the following Axioms, because they may, perhaps, be of fome Ufe hereafter.

First, If from the Sum of any two Quantities be taken either of them, the Remainder will be equal to the other Quantity: Thus, if from the Sum of any two Quantities, a+b, be taken one of them, b, there must remain the other, a.

49. Axiom 2. If from the greater of any two Quantities their Difference be taken, the Remainder must be the leffer Quantity: Thus, if a denote any Quantity, and a 4d another; if from the greater a+d be taken their Difference d, there will remain 6, the leffer Quantity.

50. Axiom 3. If to the leffer of any two Quantities be added their Difference, the Sum will be the greater Quantity: Thus, if a denote any Quantity, and ad a greater Quantity, their Difference is d; and if to a, the leffer Quantity, we add d, their Difference, the Sum will be a+d, the greater Quantity.

CHAP.

51.

CHA P. V.

Of MULTIPLICATION.

ULTIPLICATION (Multiplicatio, Lat.) is the Method of adding a Number, a given Number of Times, to itself; or, repeating it fo many Times as the Number, by which you are to multiply, contains Units. Or, the finding a Number which fhall contain any given Number, a given Number of Times.

52. The Number to be multiplied (or added a given Number of Times to itself) is called the Multiplicand (Multiplicandus, Lat.)

53. The Number by which we multiply, (viz. the Number which denotes how many Times the Number to be multiplied must be taken) is called the Multiplier (from Multiply, from Multiplico, Lat.)

54. The Sum of all thefe Additions, or the Number fo repeated, is called by Arithmeticians the Product (Productus, Lat.) and by Geometricians the Rectangle (Rectangle, Fr. Rectangulus, Lat.) The Reason of which laft Appellation will be explained in its proper Place.

55. A Multiple (Multiplex, Lat.) is a Number produced by the Multiplication of two other Numbers (each greater than an Unit.) Thus 6 is á Multiple of 2 and 3, for 3 twice repeated is 6. Or, in Euclid's Words, a Multiple is a greater Number compared with a leffer, when the leffer meafures the greater.

56. An Axiom. If equal Things be multiplied by equal Things, the Products will be equal.

57. Any two Digits may be multiplied together by adding the Multiplicand to itself, fo many Times as there are Units in the Multiplier; for Inftance, the Product of 5 by 4 may be found by repeating 5 four Times, viz. 5+5+5+5=20. By this Method the following TABLE (called the Multiplication,