EPSOM SALT. 



EQUATION, BINOMIAL. 



914 



which is more than compensated for when the charge is small by the 

 greater extent of surface exposed by the small grains. [GUNPOWDER.] 



EPSOM SALT. [MAGNESIUM, Sulphate of.] 



EQUAL. Two magnitudes are equal when one of them may Ije 

 made to coincide with the other. This is the geometrical definition of 

 Euclid, and is placed by him among the axioms, though in reality it is 

 nothing more than the definition of the word equal. Nor is it quite 

 sufficient : a triangle, for instance, aud a parallelogram, may be equal 

 in area, and yet neither can, without alteration of form, be made to 

 occupy the same space as the other. The truth is, that the idea of 

 equality is one which will admit no definition ; and moreover, it is to 

 the more general notion of the existence of ratio (of which equality is 

 one particular case) that all discussion upon this term should be 

 referred. Nor is the notion of equality to be confined to magnitudes 

 connected with space. 



Some geometers have proposed to use the word equivalent as applied 

 to surfaces of equal area but different forms ; and the distinction is at 

 least harmless. We cannot say more ; for when it is once established 

 that the term used by Euclid is to be understood in a wider sense 

 than the words of the axiom will bear, no liability to confusion 

 remains. 



There is one little warning which beginners in mathematics often 

 want. Eqital is an adjective which, by its own meaning, has no sin- 

 gular number except in conjunction with to. Thus A and B may be 

 equal ; and A may be equal to 8 ; but there is no meaning in " A is 

 equal." taken alone. 



EQUALITY, APPROACH TO. As a general rule, that which 

 may be stated as absolutely true when an equation is true, may be 

 stated as nearly true when that equation is only nearly true. Usage 

 of words however is apt to lead to mistake when it is equality, and 

 nearness to equality, which are in question : A and B are absolutely 

 equal when either of the following equations is true; one of them 

 being of course a consequence of the other; 



A 



A B = -=1 



and it is usual to say that a small quantity in nearly nothing or near to 

 nothing. In strictness, we might as well say that a large quantity is 

 near to infinity, as that a small quantity is near to nothing : both 

 infinity and nothing are limiting terms, except only as to the latter, 

 when obtained by subtraction. [INFINITE.] Nevertheless we can 

 hardly hope to abolish the common idea of small quantities being next 

 to nothing : we must therefore guard those who accept this phrase- 

 ology from the mistake to which it very frequently leads. 



It is not true that quantities are necessarily nearly equal when 

 their difference is near to nothing (meaning small). If by small we 

 here understand small with respect to the quantities themselves, it is 

 true ; but not otherwise. If A B be a small fraction of A, let it 

 be m A, where m is a small fraction of unity ; then A B= m A gives 



or B and A are in the ratio of 1 m to 1, nearly that of 1 to 1. But if 

 A and B be both small, their difference ia small : and yet that 

 difference may be itself many times greater than the smaller of the 

 two quantities from which it was obtained. If the bulk of the sun be 

 unity, the earth and moon are both small fractions ; but not nearly 

 equal. When therefore we want to think of approach to equality, we 

 must rely on approach to 



A 



- = 1, not to A B = 



B 



We are here merely guarding a phrase, not explaining principles at 

 length. In the articles INFINITE ; LIMIT ; FRACTIONS, VANISHING, 

 will be found the elementary notions, proper attention to which will 

 secure the beginner from error. 



When a person, for any common purpose, speaks of a small quantity 

 as next to nothing, he compares it with the whole mass from which it 

 ia taken, and he always means by small that which is a small fraction 

 of any important quantity. The terms small and great being purely 

 relative, may have many meanings in different circumstances ; but 

 with that we have here nothing to do. Whatever small may mean, 

 we have no right to say that quantities of small difference are nearly 

 equal, except only when the relative word small may be properly used 

 in relation to the quantities themselves. 



This last caution is the more necessary to the young mathema- 

 tician, from his frequently meeting with the words small and great in 

 an absolute tense. Though this absolute use of the words is only an 

 abbreviation [INFINITE,], he may, without care, now and then forget 

 that it. : 



KI CITATION (Astronomy). The characteristic of all the heavenly 

 motions is, that they nearly follow a simple law, but not quite. The 

 small corrections which must be added to or subtracted from the 

 result* of the simple law, in order to secure accurate prediction, are 

 called equations. Thus, the moon moves round the earth with a 

 motion which is not very far from uniform ; the average motion is 



ARTS AND SCI. DIV. VOL. m. 



therefore ascertained, and, starting from a given epoch, at which the 

 true place is known, the longitude for that epoch is first increased by 

 the longitude which would have been described by the moon, had she 

 moved with her average motion. The result must then be altered by a 

 number of different equations, some being consequences of the elliptic 

 figure of the moon's orbit, some of the sun's attraction, &c. When all 

 these equations have been annexed, the result is the moon's longitude 

 for the time proposed. 



EQUATION (in pure mathematics), an assertion of the equality of 

 two magnitudes, represented to the eye by the symbol =. Thus 

 A = B is to be understood as a proposition, declaration, or assertion 

 that the magnitude A is equal to the magnitude B. It is not imma- 

 terial to insist upon this definition ; for beginners frequently 

 confound the notion of an equation (an assertion of equality) with the 

 idea of equality itself, and speak of two equations being equal, and of 

 one equation being greater than another. 



To treat of equations is to 'write on mathematics in general ; for 

 when two magnitudes A and B are of the same kind, A must be either 

 greater than, equal to, or less than B. The objects of mathematics 

 generally require that it should be determined (supposing A and B not 

 equal) by how much one exceeds the other : and the assertion that A 

 exceeds B, and exceeds it by M, is equivalent to the equation A = B + M. 

 The assertion of inequality ia called by continental writers an 

 inequation : aud one work (we are not aware of any other) has been 

 written on the subject ; Canard's ' Traite 1 des Inequations,' &c. 



An equation may be one of two kinds : necessarily true, whatever 

 may be the value of the symbols employed, and called identical ; or 

 true only upon the supposition of some particular value being given 

 to certain magnitudes, or of some particular relations existing. The 

 latter species are called equations of condition. Thus 



are identical equations : while 



are equations of condition ; the first requiring that a should be 6, and 

 the second that x should be either 4 or 1. Again, a + 6 = 1 is an 

 equation of condition. 



Certain equations being supposed to be true, the determination of 

 all then- consequences, that is, of all equations which follow from 

 them, is the great object of mathematical analysis. The difficulties 

 which lie in the way are of various classes, and give rise to various 

 modes of considering equations. These are so widely separatee! from 

 each other, and diverge into such different subjects, that we can here 

 do no more than point out two or three of the most remarkable 

 species of inquiries. This we shall do in articles headed by the word 

 EQUATION. 



The theory of equations is a branch of algebra which is not cor- 

 rectly named ; and which we refer to THEORY OF EQUATIONS. It is 

 the theory of rational and integral expressions, which, having grown 

 out of attempts to solve equations, and having the solution of 

 equations among its most important requirements, has gained a name 

 which does not express its full scope. 



EQUATION ANNUAL. [MooN.] 



EQUATION OF THE CENTRE. [Moos ; SUN, &c.] 



EQUATION OF THE EQUINOXES. [PRECESSION.] 



EQUATION OF TIME. [SUN.] 



EQUATION, BINOMIAL. A binomial equation is an algebraical 

 equation of two terms. Its form is therefore ax* + bx 1 " = 0, all the 

 consideration of which may easily be reduced to that of one or other 

 of the forms x* _a* =0 (n being integer). All that is necessary will 

 be given in the article ROOT, and we shall here confine ourselves to 

 such statement of the mode of resolving x* + a* into factors as may 

 be useful for reference. 



First, as to x* a" . Of this x a is always a factor, and if n be 



even, x + a is also a factor. The remaining factors are all of the form 



x 1 2 cos S, 



2ir 

 in which may be either the quantity or any multiple of it less 



than 3)1. Thus whether n be 10 or 9, we go as far as four times 2*H-, 

 since 4 is the integer next less than the half either of 10 or 9. 



2* 

 According, if =v, and if it be the integer next less than the half 



of n, we have 



.c" a" ~(x a) 



x (x + a, if be even) 

 x (x- 2 cos v. ax-ra?) 

 x (a; 1 2 cos 'lv. njr + ot 2 ) 

 x ....... 



up to x (,r 3 2 cos lev. ax + a''). 



Next, as to a + o" . Of this x + a is a factor if be odd : and the 

 remaining factors are of the above form, being - or any odd multiple 



of it which does not equal n. Thus whether n be 10 or 11, we go as 

 far as times JT-*-;Z, since 9 is ths last odd integer which does not 



S n 



