113 



COMPOSITION. 



COMPUTATION. 



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given point, and losses southward, we could immediately make it a 

 necessary consequence that the balance, if any, is represented by a line 

 northward or southward, according as it is for or against. But draw a 

 line eastward, and it will readily be admitted that such line will not 

 present itself in any necessary connection with a sum lost or gained, 

 or neither lost nor gamed. For if the latter, why should a line eastward 

 be preferred to a line westward, or in any other direction ? 



An immense number of modes of composition will readily suggest 

 themselves, in which addition and subtraction are the processes by 

 which composition takes place. If I go three miles northward, and 

 then two miles farther, I go in all 3 + 2, or 5 miles northward. Other 

 modes, as in the instance first given, will suggest themselves, in which 

 multiplication and division are the compounding processes, and so on 

 ad infinitum. These are all cases in which magnitude only is con- 

 cerned ; but whenever we have both magnitude and direction, it is 

 plain that we have now both magnitude and direction to consider in 

 the effect. If I go a mile northward and then a mile eastward, the 

 whole effect, as to direction, will be, that I go to_the north-east ; as to 

 magnitude, that I go not two miles, but only / 2 miles, or 1--414 miles 

 very nearly. Here is an instance in which the components are repre- 

 sented in magnitude and direction by two sides of a triangle, while the 

 total effect is similarly represented by the third side. In the article 

 CENTRE will be found various instances in which the meaning of that 

 term implies the point at which a single action must take place, which 

 will produce the same effect as a number of different actions produced 

 on a number of points. 



In mechanics, we have to consider the combined effects of different 

 velocities, pressures, momenta, rotations, &c., communicated at the 

 same moment of time to the same body. In all, the law of composition 

 is found to be as follows : 



In what sense soever the actions at P can be represented in magnitude 

 and direction by P A and p B, in that same sense can the joint effect be 

 represented by P c, the diagonal of the parallelogram, in both magni- 

 tude and direction : or p A and A c being the actions (A c being equal 

 to P B in magnitude and direction), P c, the third side of the triangle, 

 is the united action. Thus, if at the same instant we communicate 

 motion to r in the directions P A and p B, with velocities p A and p B 

 per second, we thereby merely communicate to p a velocity p c per 

 second, in the direction P c. The same holds of momenta and pres- 

 sures; and even if we give P two separate rotations, which would 

 separately carry it round the axes P A and p B in angles per second 

 which are in the same proportion as p A and p B, the joint effect is a 

 rotation round the axis P c with an angle per second which is to the 

 angle of P A (or p B) as P c is to p A (or P B). 



We have here not to prove these things, but only to illustrate the 

 word composition. But this we must remark, that our preconceived 

 notions will never allow us to say that A is the effect of p and Q, and B 

 uf R and s, unless the application of P, Q, R, and s together will be the 

 same in effect as that of A and B together. We shall show that this 

 necessary condition of our notions of cause and effect is preserved 

 in the method of composition just described. Let p x, B y, and c z be 

 parallel to each other ; then, if our law of composition be general, 

 p B is the effect of p X and p Y. Therefore p x, p y, and p A should be 

 together equivalent to p c. But A z is equal to p Y, and P z is therefore 

 i lent to PA and P Y. Therefore AC should be the effect of PZ 

 and p x, which we immediately see it is, being as much the diagonal of 

 p X c z as it w of p B c A. 



Aa another, and a very curious instance of composition, we shall 

 notice the following : Suppose x and y are to be measured, and both 

 are subject to error, every error entailing loss in proportion to its 

 magnitude, and causing equal loss, whether it be an error of excess or 

 an error of defect. Suppose also that the errors are of such a kind that 

 the average of any number of measurement is more probably right 

 than any other. Let a and 6 be the sums which rrwould be equitable 

 to pay for insuring x and y, that is, which should be given to any one 

 who would agree to bear the loss on x and ?/ separately. The sum 

 which should then be given to one who would bear the total loss 

 arising from the possible error in .< + y is not a + b, as might at first 

 appear, but -/a' + V, or the hypothenuse of a right-angled triangle of 

 which a and 6 are as the sides. 



Our limit* will not allow us to enlarge on this subject ; we shall add 

 the two following remarks. 



1. The fact of the law of composition being the same both for 

 velocities and pressures, has caused many writers on mechanics to 

 confound the two, as if the one proved the other, which is neither true, 

 nor eyen tery probable. And other writers on mechanics, while 

 proving this general law, that actions which can be represented by the 

 two sides of a triangle produce an action which can be represented by 

 ARTS AND 8CI. DIV. VOL. III. 



;he third side, have restricted the proposition, and seem to imagine 

 ;hat what they prove is true of forces only. This, with great deference 

 ;o such a writer, we conceive M. Poisson to have done in the well- 

 cnown proof at the beginning of his mechanics, a work which we may 

 ;ake this opportunity of saying, we hold in higher estimation than any 

 other elementary mathematico-physical work whatsoever. 



2. The difficulties of negative quantities in algebra arose from a want 

 of generality, which gave rise to the attempt to express composition by 

 addition only or by subtraction only, where either addition or sub- 

 traction might be requisite ; and the difficulties of impossible quanti- 

 ties arose out of a similar deficiency, bearing the most complete 

 analogy to trying to compound in magnitude only, in cases where 

 both diversity of magnitude and of direction should have been con- 

 sidered. 



COMPOST. [MANURE.] 



COMPOUND, that which results from composition. [COMPOSI- 

 TION.] 



COMPOUND ADDITION. [ADDITION.] 



COMPOUND INTEREST. [INTEREST.] 



COMPOUND QUANTITIES [ARITHMETIC], quantities in which 

 more than one unit is employed, as in 2 pounds, 3 shillings, and 6 

 pence : 2 miles, 3 yards, and 4 inches. 



COMPOUND RADICALS. A term applied in chemistry to those 

 combinations of elements which act towards oxygen, hydrogen, and 

 acids, as simple elements. [ORGANIC RADICALS.] 



COMPOUND RATIO. [COMPOSITION; RATIO.] 



COMPOUNDING OF A FELONY. [FELONY.] 



COMPRESSIBILITY OF WATER. [ELASTICITY.] 



COMPURGATOR. In the middle ages a practice prevailed, 

 derived from the canon law, of permitting persons accused of certain 

 crimes to clear themselves by purgation. In these cases the accused 

 party formally swore to his innocence, and, in corroboration of his 

 oath, twelve other persons, who knew him, swore that they believed 

 in their consciences that he stated the truth. These twelve persons 

 were called compurgators. (Ducange, ' Juramentum.') This proceed- 

 ing appears to have existed among the Saxons, and, in process of time, 

 it came into use in England in civil cases of simple contract debts. 

 [WAGER OF LAW.] The ceremony of canonical purgation of clerks- 

 convict, which was nothing more than the formal oath of the party 

 accused, and the oaths of his twelve compurgators, continued in 

 England until it was abolished by the stat. 18 Eliz. c. 7. [BENEFIT 

 OF CLERGY.] (Blackst ' Comrn.,' Dr. Kerr's edit., vol. iii., 364 ; iv., 

 434.) 



COMPUTATION. We need not tell those who are acquainted with 

 the existing treatises on arithmetic, that in no one instance do they 

 pretend to give any mode of forming good habits of computation. 

 The beginner, after receiving instructions as to what is to be done in 

 the several great rules of arithmetic, is allowed to manage the details 

 as he can. 



The mere mechanical art of computation, apart from arithmetical 

 reasoning or application to subjects of interest, is no very lofty exer- 

 cise of the mind. A wonderful degree qf proficiency in it can be 

 attained by many who find connected reasoning almost an impossibility ; 

 and on the other hand, some of the first among mathematical dis- 

 coverers have hardly arrived at more than the expertness of an ordinary 

 schoolboy. It is one of those arts among many which are accessible 

 to all who begin with a determination to conquer difficulties, and a 

 power of arriving at methodical habits : no person who, after begin- 

 ning in the right way, is obliged to confess a total failure, has any 

 ground to suppose that he could master a common manual art: he 

 may be a genius, but nothing could make him a weaver. That we 

 may not frighten any one of the thousands who are miserable computers 

 after going through years of school discipline, and whose minds are 

 too well made to allow them to flatter themselves that they were 

 above it, it is but fan- to say that very few are allowed to begin in 

 the right way. Every merely mechanical business must be learned 

 by a sufficient repetition of the most purely elementary steps. A 

 discipline of the mind may be taken up at the wrong place, and still 

 be a discipline, though not so perfect as it might have been. But in 

 what is merely art, nothing can compensate for the want of habit of 

 operation duly learned at the proper time. Now computation is only 

 an art : its elements are a small number of acts of memory : its details 

 consist in a still smaller number of operations, each of which, by 

 itself, is of the utmost simplicity. 



Many readers will suppose us, in speaking of the elementary rules 

 of arithmetic, to mean addition, subtraction, multiplication, and division, 

 as given in the books : but we should as soon think of saying that 

 the elementary operations of a journeyman tailor's business are the 

 making of coats, waistcoats, and trowsers. The rules just named are 

 the perfection of computation, not its commencement : he who can 

 do them all with ease and accuracy is a calculator. The fundamental 

 operations of which we speak are to those elaborate processes just 

 what threading a needle and drawing a stitch are to the making of a 

 coat. We can carry the comparison still further : and its justice is 

 not accidental, but the necessary conseqjusnce of the resemblance of 

 all mechanical operations We do not say that a workman who is 

 capable of joining two pieces of cloth together with streu^tli and neat- 

 ness is a finished tailor : he cannot therefore choose cloth, cut out 



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