In the last example, and all others, where fractional and radical quantities are concerned, every fuch quantity, exclufive of its coefficient, is to be treated in all refpects like a fimple quantity expreffed by a fingle letter. 3°. When in the quantities to be added, there are Terms without others like to them, write them down with their proper figns. Thus a+26 added to 3+ d makes a+2b+3c+d. And `aat bb added to a+b makes aa+bb+a+b. Here follow a few examples for the Learner's exer cife, wherein all the three foregoing rules take place promiscuously. 1.: 2aa + 3ab8cf + d3 Sum 5aa +16cc + d2 = d2 + 30. - of quantities together, will be equal to the fum of all the affirmative Terms diminished by the fum of all the negative ones (confidered independent of their figns); from whence the reafon of the fecond general Rule is apparent. As to the cafe where the quantities are unlike, it is plain that fuch quantities cannot be united into one, or otherwife added, than by their figns; thus, for example, let a be fuppoféd to reprefent a Crown, and ba Shilling; then the fum of a and b can be neither 2a nor 2b, that is, neither two crowns nor two fhillings, but one crown plus one fhilling, of a+b. 3.5 2... 7 Vaa — xx + 5 ax 8 √aa-xx + 12 √aa + 4xx 8 Vax + 15 Vaa-xx 8 √ aa + 4xx 10 √aa + 4xx Sum 13a2 + 22ab + 3b3 + a3- ·c3 + 20. bc. SECTION. III. Of Subtraction. Subtraction, in Algebra, is performed by changing all the Signs of the Subtrahend (or conceiving them to be changed) and then connecting the quantities, as in addition. In the fecond example, conceiving the figns of the fubtrahend to be changed to their contrary, that of 36 becomes +; and fo the figns of 36 and 56 being alike, the coefficients 3 and 5 are to be added together, by cafe I of addition. The fame thing happens in the third example; fince the fign of 3b, when changed, is and therefore the fame with that of 55. But, in the fourth example, the figns of 36 and 5b, after that of 3b is changed, being unlike, the difference of the coefficients muit be taken conformable to case 2 in addition. Other Other examples in Subtraction, may be as follow: From 20ax+ 5bc 7aa From qV ax + g√ by 5aa take take 12ax - 5V ax + 12V by Rem. 12 ax- ·3 by• Rem.-3 V aa xx ·† 25√ a3 — x3 + 2 V aa In this laft example the quantity a" in the fubtrahend, being without à coefficient, an unit is to be underftood; for 1a and a mean the fame thing. The like is to be observed in all other fimilar cafes. The Grounds of the general rule for the fubtraction of algebraic quantities may be explained thus: Let it be here required to fubtract 5a-3h from 8a+ 5b (as in ex. 2.) It is plain, in the firft place, that, if the affirmative part 5a were alone to be fubtracted, the remainder would then be 8a+ 5b5a; but, as the quantity actually propofed to be fubtracted is less than 5a by 3b, too much has been taken away by 3b; and therefore the true remainder will be greater than 8a+ 5b5a by 3b; and fo will be truly expreffed by 8a+ 56-5a+3b: wherein / the figns of the two laft terms are both contrary to what they were given in the fubtrahend; and where the whole, by uniting the like terms, is reduced to 3a+8b, as in the example. SECTION IV. Of Multiplication. EFORE I proceed to lay down the neceffary rules for multiplying quantities one by another, it may be proper to premife the following particulars, in order to give the Learner a clear idea of the reason and certainty of fuch rules. First, then, it is to be obferved, that when several quantities are to be multiplied continually together, the refult, or product, will come out exactly the fame, multiply them according to what order you will. Thus a x b xc, a x c x b, bx c xa, &c. have all the fame value, and may be used indifferently: To illuftrate which we may suppose a = 2, b = 3, and c = 4; then will a x bx c = 2 × 3 × 4 = 24; axcxb=2×4 × 3 = 24; and bx c xa = 3 X 4 X 2 = 24. Secondly. If any number of quantities be multiplied continually together, and any other number of quantities be alfo multiplied continually together, and then the two products one into the other, the quantity thence arifing will be equal to the quantity that arifes by multiplying all the propofed quantities continually together. Thus will abc × de = a x b x cxdxe; fo that, if a was 2, b = 3, c = 4, d = 5, e = 6, then would abc × de = 24 × 30 =720, and a x b x c x d x e = 2 × 3 × 4 × 5 ×6= 720. The general Demonftration of thefe obfervations is given below in the notes. The following Demonftrations depend on this Prin ciple, that if two quantities, whereof the one is n times as great as the other (n being any number at pleasure), be multiplied by one and the fame quantity, the product, in the one cafe, will also be n times as great as in the other. The greater quantity may be conceived to be divided into n parts, equal, each, to the leffer quantity; and the product of each part (by the given multiplier) will be The multiplication of algebraic quantities may be confidered in the seven following cafes. be equal to that of the faid leffer quantity; therefore the fum of the products of all the parts, which make up the whole greater product, muft neceffarily be n times as great as the leffer product, or the product of one single part, alone. This being premised, it will readily appear, in the first place, that bx a and a x b are equal to each other: For, bx a being 6 times as great as I xa (because the multiplicand is b times as great) it must therefore be equal to Ix a (or a), repeated b times, that is, equal to a xb, by the definition of multiplication. In the fame manner, the equality of all the variations, or products, abc, bâc, ach, cab, bca, cba (where the number of factors is 3) may be inferred: For those that have the laft factors the fame (which I call of the fame class) are manifeftly equal, being produced of equal quantities multiplied by the fame quantity: And, to be fatisfied that thofe of different claffes, as abc and acb, are likewife equal, we need only confider, that, fince ac x b1 is c times as great as a x b (because the multiplicand is times as great) it must therefore be equal to a × b taken e times, that is, equal to a × b × c, by the definition of multiplication. Univerfally. If all the Products, when the number of factors is n, be equal, all the Products, when the number of factors is n + 1, will likewife be equal: For thofe of the fame clafs are equal, being produced of equal quantities multiplied by the fame quantity: and, to fhew that thofe of different claffes are equal alfo, we need only take two Products which differ in their two laft factors, and have all the preceding ones according to the fame order, and prove them to b☛ equal. These two factors we will fuppofe to be reprefented by r and s, and the Product of all the preceding ones by p; then the two Products themselves will be represented by prs and psr, which are equal, by cafe 2. Thus, |