ADD 



ADD 



bfrs, er quantitifs, fcverally added one to anoilier. Or, 

 arliruion is the invention of a number, from two or more 

 homogeneous ones given, which is equal to the given num- 

 bers taken jointly, or together. 



The number thus found is called the/ww, or aggregate 

 of the numbers given. 



The character of aiMilion is +, which we ufuaily cxprefs 

 by plus. Thus 3+4 denotes the fum of 3 and 4 ; and is 

 read 3 plus 4. 



The addition of finiple numbers is eafy. Thus it is rea- 

 dily perceived that 7 and 9, or 7 -j- 9, make 16; and 

 II -|- 15, make 2rt. 



In longer, or compounded nun\bcrs, the buliiicfs is per- 

 formed by writing the given numbers in a row downwards ; 

 homogeneous under homogeneous, /. c. units under units, 

 tens under tens, &c. and accurately collecling the fums of 

 the refpeClive columns. 



To do this, we begin at the bottom of the outmoft row 

 or column to the riglit ; and if the amount of this column 

 be ten, or fome number of tens, we ftt down only the over- 

 plus, and carry one for each ten to the next column. 



Suppofe, e.g. the numbers 1357 and 172 were given to 

 be added : write either of them v. gr. 172, under the other 

 13^7 ; fo that the units of the one, viz. 2, ftand 

 under the units of the other, viz. 7 ; and the other 

 numbers of the one, under the correfpondent ones of 

 the other, viz. the place of tens under tens, as 7 

 under 5 ; and that of hundreds, viz. i, under the 

 place of hundreds of the other, 3. Then, begin- 

 ning, fay 2 and 7 make 9 ; which write underneath ; alfo 

 7 and 5 make 12 ; the lalt of which two numbers, viz. 2, 

 is to be written, and the other one referved in your mind to 

 be added to the next row, 1 and 3 : then fay i and i make 

 2, which added to 3 make 5 : this written underneath, and 

 there will remain only i, the firft figure of the upper row of 

 numbers, which alfo muft be written underneath ; and thus 

 you have the whole fum, viz. 1529. The fame method 

 will extend to any number of fums, which are required to 

 be united in one. 



When a great number of feparate fums, or numbers, are 

 to be added, it is more eafy to feparate them into two or 

 more parcels, which may be added ftparately, and then their 

 fums added together for the total amount : and thus, by 

 dividing the numbers into parcels in different ways, the 

 truth of the addition may be proved. 



Another method of proving addition was fuggefted by 

 Dr. Wallis in his arithmetic, publifhed in 1657, by calling 

 out the nines. Thus, add the figures of each line of num- 

 bers together fcparately, and call out always 9 from the 

 fums as they arife, adding the overplus to the next figure, 

 and fetting down at the end of each line the excefs above 

 the nine or nines. Purfue the fame procefs with the fum 

 total, and the former exceffes of 9, and the laft exceflies 

 will be equal when the work is right. The former examples 

 may be thus proved : 



1357 

 172 



1529 



1529 



o 



u 

 y. 

 M 



Thus alfo ; 



350709 



3 1 806500 



339087 



4601 1 



2935 



32545242 



6 



^ 5 

 'B 3 



"^ 3 



W — 



Addition of numbers of different denominations, for in- 

 flance, of pounds, (hillings, and pence, or yards, feet, and 

 inches, is performed by adding or fumming up each deno- 

 mination by itfelf, always beginning with the loweft ; and 

 if, after the addition, there be enough to make one of the 



next higher denomination, for inftance, pence enough to 

 make one or more (liiUings, or inches to make one or more 

 feet, they mull be added to the figures of that deno- 

 mination, that is, to the fhillings or feet, only referving 

 the odd remaining pence or inches to be put down in the 

 place of pence or inches. And the fame rule is to be ob- 

 fervcd of (billings with regard to pounds ; and of feet w ilh 

 regard to yards. 



As in the following examples : 



£■ 

 120 



65 

 9 



'5 

 12 



8 



9 

 5 



yds, 



271 



36 



14 



10 

 2 

 2 



Avoirdupois weight. 

 dr. 



195 16 2 Sum 326 



Ih. 

 15 



4 



12 



o 



OS. 



I I 



10 

 o 



15 



12 

 O 



'3 

 9 



Sum 33 6 2 



Addition of Decimals. Sec Decimal. 



Addition of f^iil^ar Fratlioiis. See Fraction. 



Addition if Logarithms. See Logarithm. 



Addition of Ratios is ufed by fome authors in the fame 

 fenfe with composition of ratios, which fee. 



Addition, in Algebra, or the addition of indeterminate 

 quantities, expreffcd by letters of the alphabet, is performed 

 by conneAing the quantities to be added, by their proper 

 figns ; and alto by uniting into one fum, thofe that can be 

 fo united ; /. e. fimilar quantities, by adding their co-effici- 

 ents together if they have the fame figns, or fubtracling 

 thofe which have diflerent figns. So that addition compre- 

 hends three cafes. 



Cafe I. To add quantities which are like, with hke figns : 

 add all the co-efficients together, and to their fum annex the 

 common quantities, and prefix the common fign. Thus, 



7fl-}-9<7=i6a. And iibc+i^bc=.26bc. Alfo 32.-^-5 " 



el ^ ^ 



=8-: and 2^ ac-{--]^ac=g^ ac; and 6v^ai — .-vx + 



'JVab — .\x=:i^'i/ab — x.v: and in like manner, 6^/3+7 

 y/3 = I3v^3. Again, a,^/iic-{-b^ ac = a-\-b\^ac ; and 



2a-\-i,c'/ ^ax'- 



^aV ^a : 



a-\-x a-\-x 



And 6a-j-9i— 3r — 4 

 4a4-5^-2f-3 



Sa + j cV ■jax '- 

 a+x 



\oa-\-l^~ S<'~1 



Cafe II. To add quantities which are hke with unlike figns: 



add all the affirmative or pofitive co-efficients into one fum, and 



all the negative ones into another ; then fubtraft the leall of 



thefe fums from the greateft, and to the difference prefix the 



fign of the greateft, and annex the common quantity. 



I , A.axiiax. 



ihus, —2 and— 3 make —5; r-and ; — make 



^ b b 



^— ^; ~a \/ a x and —b^^ax make —a — bv a x. Alfo, 



3 — 2 = 1; ga — -]a=2a; 



— — —J-; and —a^Uc 



-\-b^/ac=b — a,,/ac; and, 2 — 3 = —! ; 



b 



I I (7.V 



4(7 .V 



7«X 



and 2,J ac — •] ,^ ac-= -—^^/ac. 



b-^ b - 

 Again, -3,7-f. 



7a-{-8a — ^— 2a = — 6^4-15^ — -l-Qa ; and —^xy — ^ xy 

 •\-^xy-\-']xy=:—ixy-\-l$xy=-\--jXy; and —6^ax-\-Z 



V'ux — 5^ax-f-ioy'fl.v= —II y^a.v-j- 12^ ax = ,^ ax: 

 And ^a■\-^^b-%c-^ 

 — 6a-3i4-4f4-4 



2.7-1-4^-4^-3 • 



Cafe 



