a 4' + 20 20 The Reason of the Operations in this case may be easily under food by any one that duly considers the comparing of Stock and Debts together, or the Ballancing of Accounts betwixt Debtor and Creditor. That is, the Affirmative Quantities represent the Stock or Creditor : The Negative Quantities represent the Debts; and their Sum represents the Ballance, &c. Cafe 3. When the Quantities are unlike, set them all down, without altering their Signs; and thence will arise compound Quantities, which can be no otherwise added but by their Signs. Thuslila s6+ 7 dc 2.6 6 1 +232 +6 a-6156+ 7 di + 4 à Here follow a few Examples wherein all the 3 Gafes are promiscuously concerned. 11 aa-t-2ab+bbl. 8 ab tbc-37 -7 ab--be + 42-50 i+23 aa- 2 ab tabbl ab-t 5-60 Iaa -2 ab +-bb3bc + ab- 45 +4ab-+6040 6 bi-- ab + da 1+213!2a + 2ab + bb 3 he + 40 salatb-ab -791īca -d 3a +4abc-bb + 30 2 abc-25 dd + 2 aa 3abc-3 1.7.2 +3141-to dd to 2aa + bb 2 -4ab 2 45 t. da 2 2 • abc + 2 Sect. 2. Subtraction of whole Duantities. SUBTRACTION of whole Quantities is performed by one general Rule. RU L E. Change all the signs of the Subtrahend, (viz. of those Quantities which are to be subtracted) or Juppose them in your Mind to be changed; then add all the Quantities together, as before in Addition, and their Sum will be tbe true Remainder or Difference required. This general Rule is deduced from these evident Truths. To subtract an Affirmative Quantity, from an Affirmative, is the same as to add a Negative Quantity to an Affirmative: that is +2a taken from + 3a, is the same with ~ 2a added to +39. · Consequently, to subtract a Negative Quantity from an Affire mative, will be the same as to add an Affirmative Quantity to an Affirmative; that is - 2a taken from +3a will be the same with +20 added to +3a. Exam.1. Exam.2. Exam.3. Exam.4- 15bc 36 8bc I-2 56 7bc Exam. 5. Exam. 6. | Exam. 7. 1150 + 12b15a 12 6 9 ab + 36 (2/2a + 762a 703ab + 24 1313a + 5b)3 a 5616 ab + 12 Exam.8 Exam.9. Exam. 10. Exam. 11. 2 a 2 abd 6bc + 7 abd 1-231+ 5a · 5 a + 7 bc 3а 9 abd If these 13 Examples be compared with those in Addition, the Work will appear very evident, these being only the Converse or Proof of those; according to the Nature of Addition and Subtraction in common Aritbmetick. More Examples in Subtraction. 12 20 C+13 a b 24-46 -2.3130 +13 - 3a + bla 20 +46 1 lat b.- 54 76 Ild3b-bc-75 a-b5dt 70 1113a +46 +bc+210176 a+b+5d-70 That ab taken from a tb leaves + 2b for the Remainder, as in the first of these Examples, may be thus proved : Let'|a+b= And 24 2763 a%3Dx+6 . per Axiom 1. 1-3 41 b =Z-*-b per Axiom 2. 4+615'2b = -x which was to be proved. The Truth of all Operations in Subtraction, where any Doubt arises, may be proved, by adding the Subtrahend to the Remainder, as in Common Arithmetick. E X A M P L E. From 11+52 o -9bc Take 2 - 2a +36 6 da Subtrahend. 12 31+7a 361 + 6da9bc Remainder. 2+3+1+52 9bc Prouf. Sect. 3. Pultiplication of whole Duantities. ULTIPLICATION of whole Quantities admits Case 1. When the Quantities have like Signs, and no Coëfficients, set or join them together, and prefix the Sign + before them; and that will be their product. Exam.z. Exam.z. Exam.3. Exam. 4. a+b -b 216 -b12 tab ladybdit adtbd Thus { I x2 3 lab Cafe 2. If there be Coefficients; multiply them, and to their Product adjoin the Quantities set together as before. Thus us { 2 76 Thus { {: Exam.5. Exam.6. Exam. 7. Exam. 8. -6d Thus 3 atab atb 36 6 56 18 a +1261 sab + 5bb Cafe 3. When the Quantities have unlike Signs; join them and the Product of their Coëfficients together (as before) but prefix the Sign -- before them ; Exam.9. Exam. 10. Exam. u. Exam. 12. 6d 4 amb 31 in But + into —, or - First to prove that -76 into +3f=-21bf. As in Ex. 11. per Axiom 1. 2 x 3/412af=zibf per Axiom 3. 4-21 bf|5112 af-216f=0 per Axiom 2. Consequently + into —, or - into + produces , which was the Thing to be proved. Secondly to prove that — 7 b into - 3f gives + 21 bf as in 76 Or Thus, fuppose {i=19} and { Or these may be otherwise proved by Numbers. {ale c=12] = 8 bers. Then 4 b6 id=4 per Axiom z. Consequently, a-bxcd=6*4= 24, per Axiom 3. but -bxc-d, according to the precedent Rules, will be, as-cb+bd-da, which if true must be equal to 24. Proof a c= 20 x 12 = 240 cb=12 X 14 = 168 da= 8 x 20 = 160 Hence actbd=352 per Axiom 1. And cb+da= 328 which being subtracted, Leaves actbd-cb-da=352-328=24, which plainly thews, That + into in the Product. And into produces + 2: E. D. Note, If the Multiplier consists of several Terms, then every one of those Terms must be multiplied into all the Terms of the Multiplicand; and the Sum of those particular Products, will be the Product required, as in Common Arithmetick. Proof?bd=14x 8=112 } E X A M P L E S. 2 atb-d 7b75d -b 13 a5f Ixa 3 a atbada 21 ba+15da 1 x6/4 -ba-bbdb -35bf-25df 3+41512a-da-66+db21b2+15 de-35bf — 25 df a amba ma a tb 25 - 30 」二 32-4b aa+za+4 aa-ba tbb a+b 2 2 +baa-bbat bbb 1 x 23 aa a-8 laaabbb 40-8 Scat. |