| Show PHE HE THAT IS IGNORANT OF SOBERS IS SCARCE HALF A MAN uan CHARLES XII OF w 1 I 41 an in the present attempt to convey to t the minds tf cf some of your readers a slight knowledge of mathematical science both as regards measurements by numbers arithmetic a anti antl 1 ed measurements by d mansions men siong geometry t the the sketch I 1 offer of each is nece harily brief tand ara arn Jia perfect but my end will b be gained if I 1 afford that amount of information on the sub jf ast t which ii generally possessed by persons persona of of moderately well cultivated intellect ga awa reco recognition of the value of 0 numbers is ajr bye witti with the dawn of mental r cultivation in y every community but considerable progress lindst must be made before methods of reckoning reck aning jh olire ollre are reduced to a regular system and a nota tion adopted to express large jarge aud and complex quantities ii titles tad ted add zdi ari inability to reckon bey nd a few num berb deas gets bers is always a proof of mental obscurity alid aud attl aati rein sein this state various savage savage nations have b en discovered by travelers some are found to be able to count as far as five the digits of tb e hied most likely familiarizing them with tria tar all aia umber number 1 but any further quantity is r rather a ahw said t to 0 consist of so many fives or is expressed by the more convenient phrase a great many 11 among our oer various indian tribes any great number which the mind is incapable of distinctly recognizing and naming is figuratively described by comparing it to the leaves of the forest and in the same manner the untutored negro negro of africa would define any quantity of a avast vast arrouet by pointing to a handful of sand of the desert in inthe the first advance of any early people toward civilization it would be found imps bible ible B to give a separate name to each separate number which they had occasion occa lon ion to describe it would thero therefore fore be necessary to consider large numbers as only multiplication of certain jorn baen smaller aller ailer ones and to name them thed ig feigly I 1 Y 9 this is no doubt what cave eave rise to classes of 0 numbers which are different in different countries for instance be the chinese count by tl t wo n 03 i s the ancient mexicans reckon by fa tours fours u r a some count by fives the me hebrews from froman an early period reckoned by tens the greeks greeks adopted this plan from the greeks it came to the romans and by them was spread aver dver over a large part of the world the hebrew improved rn proved and grecian and roman numbers were perhaps sufficient to express any s agle nimber xie the the greeks certainly co to overcome many abs obstacles acles in the budness business ot of calculation and abd even could express fractions in fact tact ahe dhe romans were obliged where mental calculation cu lation would not serve to resort to a me hanibal V Ha a process for performing problems in arl art li A box ot of pebbles called lo iulus tulus yndia india ada board called abacus constituted their means eans of calculation and of these every schoolboy and many other persons possessed a set the word calculate claims no higher deacent descent than from calculus a stone or pebble tae labor tabor of counting and arranging the pebbles was greatly abridged b by y dr drawing awin across the board a horizontal line above which each single pebble had the power of five and afterward the whole system was made more congenie con venie t by substituting beads leads strung on parallel threads or peg pegs stuck in grooves methods of calculating still used in russia and china and found con lenient in certain departments of the ro roman man catholic devotion and in several familiar games of more civilized countries the numbers now in uee use and the ade of causing them by a peculiar situation to express any number and thereby the processes of arithmetic have been ren alered so highly conven convenient ien len have heretofore supposed to be of indian ori orill origin il transmitted transmit td d tb through the per blans to the arabs and by them therk introduced into europe in the tenth century when the moors invaded and became maser of spain in the elevens ele eie venn veno century Gerber geibert ger Gei bert bett tp a benedictine monk of fleury feury and who afterward ascended the papal throne under the designation of Sylv sylvester ester eiter the second traveled into spain and studied died for several years the science there cultivated by the moors among other acquisitions he gained from that sik sin singular ular people a knovie kno eledge dge of what are now called trailed thie toe arabic numerals Numer alsand alband and of the mode of arithmetic founded on them which he forthwith disclosed to the chrls chris chrisian ian world by ishom whom at first his bis kearnin learning caused him to be accused of an alliance with evil ic if would be impossible to calculate even by their own transcend transcendent ant power the service wb wh ch they have rendered to mankind s the arabic numerals take the follo followings fol foi lowins winz well vdell known form 1 2 34 3 4 5 6 7 8 9 0 the last called a nought nothing or cipher 4 ia Is in reality taken by itself expressive of an absence of number or nothing but incon in connection with otha other er numbers it be becomes cordes expressive in a very remarkable manner the valuable peculiarity of af the arabie arable notation la Is uie lie the enlargement and variety of values which can cad be given to we ide fiure figures by associate n ng them enere are four elementary departs nis in arithmetic addi addition tion tiou tion til and division addition is the operation b several numbers are united in one 1 ai be number thus obtained is called the gum flum or amount B by y multiplication we ascertain what a number amounts to when repeated a iriven number ot of times hence multiplication is a short abort method of addition under certain by we ascertain how much greater one number is than another adother pr or what remal remains 11 when a less number la is taken from fram a greater oe or tb the e difference between any two numbers bumbera by division we find how many times one number is contained in another it is the converse of 0 multiplication the product and one factor being given and the other re rebutting resulting from the operation in this thia short abort essay it is not my design to enter into a complete and full detail of ofala all the different rules and regulations of arithmetic i but shall pass them over as slightly as pcs I 1 sibie bible the rule of three which has been considered by many as belp bein being 1 I of great and vital importance to the fhe ae arithmetician is simply the rule of ratio which we now call proportion j ratto ratio rati Kati katinis ois is the relation which one aka quantity n city 1 bears to another of the same kind with respect to magnitude or the ratio of two numbers is the quotient resulting from the division of the first by the s second e cond thus the ratio of f 13 to 17 is 1317 1347 13 47 17 and that of 65 to 85 is 65 63 13 85 55 17 proportion is equality of ratio four numbers are said baid to be the ratio of the first to the second is the same as the ratio I 1 of the third to the fourth hence I 1 are proportion als alb the first and fourth are called the extreme terms and the second and third are called the means it four numbers be in fro pro the product of the extremes ie is equal to that of the means when the answer to a question depends upon several cond conditions the process by which we effect t the he solution OP is called compound proportion the best illustration Is an example viz 4 sold to B 20 hogsheads hog of molasses at a loss of 7 lb ib 12 per percent cent B sold the same to 0 and gained 20 per cent C acthen then sold the whole to JO for and thereby ga ned 12 12 1 2 per ent tent how bow much did jl 4 give per hogshead 0 f arit in making this statement we reason thus since 0 by selling the whole SOLUTION for gained 12 12 12 per cent which was 18 of what 9 8 it cost him he sold if for 9 6 5 parts whereas he bought it 37 91 40 for 8 we therefore put the 8 20 on kine line the right of the line 1 and we have the statement of the 1 cost price to 0 next we ue perceive that B sold to C so 3 as to gain 20 per cent which was 15 1 5 of what it cost he sold it therefore for 6 parts whereas it cost him but 5 hence we place the 5 on the right of the line and the statement thus far will give the cost price to B nw we perceive that jl 4 sold to B sai sat at at a loss of 7 12 per cent which w wab waa as of 0 what it coat him he therefore bod sod sold BOA it tur fur 37 parts whereas it cost him 40 parts hence we place the 40 on the rig right bt of the line the rule of proportion generally given directs the learner to reduce the terms to the lowest west denoma denomination nation mentioned aed which in e effect c is tea teaching ching to express those terms by the greatest possible number of figures now the opposite of this thesis is certainly the only rational course to pur par purgie ie that is to express the terms of the tb stat statement emit by the least number of figures that the proportion will admit of to do this consider each lower denomination as a fraction of the next higher thus 5 cwt cat 3 ars and 7 cwt cat we arrive at this result by a very simple mental process thus IS 12 7 of a quarter then 3 37 3 7 quarters are 24 7 quarters and 1 hence 24 parts are 67 6 7 of 28 parts disregard ing the denominators and fia finally ally aily 5 67 6 7 cat cwt 41 7 cwt cat the pupil who now adopts the course of making his bis solutions of as leat beat gleat an extent as possible by a purely mental operation will make much greater progress than by solving his questions by the mere mechanical procesa of making figures whilst at the same time bd will lil ill thus strengthen his bis memory and ana develop the reasoning powers of 0 his hla mind more in one da day y than would woula result from the common in method e t od of P pursuing the study of this sc oc ence in a week it should be impressed on the mind ot uthe the learner that he should accustom himself to use as few ew figures in the solution as possible but after the statement is made the answer should be obtained simply by a mental operation by pursuing the course here suggested much time will be gained a great amount of useless labor dispensed with and the intellectual capacities pa cities of the learner invigorated at every stage of hia progress 1 there thele is no necessity for the special rules for the solution of questions in I 1 1 are and tret loss and gain barter interest discount etc which we find in the generally all ail used in our schools ft the pupil has baa learned the proper exert exercises ses of f his reasoning faculties he be will ascertain generally without much assistance from the teacher what is required as the answer to a question and will make hs statement and solution accordingly it la is evident that all methods of computation lies in their abrev ty hence algebra must be he considered as one of the most important departments of mathematical sc ence on account ot of th extreme rapidity and certainty with which it enables us to determine the most involved and intricate questions the term terra algebra is of arabic origin and hag has a reference to the resolution and compo of quan quantity titi fj in the manner in which it is applied it embodies a method of I 1 ing calculations bys byi means of and abbreviations which are used instead of worda words and phrases so that it may be called a system of bois bols 4 although alt AIt hough bough it is a science aa of eale cale b jet pei yet its operations must not oot be confounded with haae ot of arithmetic I 1 all calculations in refer to some particular individual question whereas those of algebra refer to a whole class oe of questions one great advant advantage ag e in algebra is that all the steps of any particular course ot of reasoning are by means of symbols placed at once before the eye so that the mind bein being unimpeded in its operations proceeds uninterrupted from one step of reasoning to another until the solution of t the h e question I 1 is a attained symbols afe arc used to represent not only the known but also the unknown uni cnown quantities the present custom is to represent the known q ian lan by the first letters of 0 the alphabet as a 0 b c etc and the unknown quantities by the last x y s r the signs sighs used in algebra are arel 1 i l X 2 3 in add tion the same process procesa is always used in algebra I 1 as in arithmetic whenever like quantities ties with like signs are to be b e added but it often happens that like quantities I 1 which bich are to be added together have unlike signs addition his in algebra a far more extended sign 11 ailon allon than arithmetic in for tx ample example to add ta i aa to 8 aa it is evident that after aa iua lua l aai i aa have been added according to the usual method aa must be bulb str acted hence the general rule as lad iad la d down by algebraists adding ot of like quani quantities ties with like signs is to add the coefficient efficient co fl cf the positive terms and the begat negative live terms the less sum to be ed from the greater and to the difference the sign ag agn of the greater must be annexed with the common letter or letters tile the multiplication is berf performed ormed by multiplying muli muil mul mui plying as in arithmetic the coefficients co ca of the quantities and then prefixing the proper signs and annexing letter in di divis vis on all letters common to both quantities must be omitted in the quotient and when the same bame letters to both with different indices the index of the letters must be ed from that of the the doctrine of equation constitutes 7 by far the most important part of algebra it being one of the principal objects of 0 mathematics to reduce a I 1 questions to the form of equation and then to ascertain the value of the unknown quantities by means of their relations to other quantities titles of which the value ia is knon san yan many I 1 any problems which are now quickly q and read ly determined by being reduced to equation used formerly to be solved by tedious and intricate arithmetical arithmetic ar ruiea rulea and may still be found in old treatises on arithmetic arranged under the title of double and single position false position allegation I 1 etc i biu Blu alfons receive different names acco according to the highest power of 0 the unknown quantities ies tes contained in them the quantities 0 of which an equation is formed or comp sed bed bed sea are called its terms and the parts that stand on the right and left of the si bin sign sin n are called the members or sides of the gua equation qua tion when it is desired to determine any question that may inay arise respecting the value of sobie bome some unknown quantity py means of an equation two distinct steps or operations are requisite the first step consists in translating the question from the colloquial language of common life into the peculiar Vc ullar analytical language of the science I 1 the second step consists in finding by given rules the answer to the question or in other words the solution to the expertness and facility in performing the former operation cannot be produced by any set bet t 0 of f rules in this ghisi as in many other processes pr ocse s practice is the best teacher every new quei quel question tion requires a new process of reasoning A quadratic equation means a square equation the term being derived from the latin quadrate quadra fus lut squared A quadratic equation therefore is merely an equation equa ion in which the unknown quantity is squared or raised to the second degree there are two binds kinds ol 01 quadra lc ic equations namely pure and i pure quadratic equations are those ia in |
