MAX 



MAX 



'{principle, in which sense it denotes much 

 the same with axiom. See AXIOM. 



Maxims are a kind of propositions, 

 which have passed for principles of sci- 

 ence, and which, being- self-evident, have 

 been by some supposed innate. 



MAXIMUM, in mathematics, denotes 

 the greatest state or quantity attainable 

 in a given case, or the greatest value of a 

 variable quantity ; hence it stands oppos- 

 ed to the minimum, which is the least 

 possible quantity in any case. Thus in the 

 expression a 1 - b x, where a and b are 

 constant, and x variable, the value of the 

 expression will increase as b x or x dimi- 

 nishes, and it will be greatest, or a maxi- 

 mum, when x is least, or =0. The ex- 



b b v 



pression a 2 increases as dimi- 



x x 



nishes, that is, as x increases, and it will 

 be a maximum when x is infinite. If along 

 the diameter, K Z (Plate X. Miscel. fig. 

 4.) of a circle, a perpendicular ordinate, 

 L M, be conceived to move from K to Z, 

 it increases till it arrive at the centre, 

 where it is greatest, and from thence it 

 decreases till it vanishes at Z. Some quan- 

 tities continually increase, and have no 

 maximum, unless what is infinite, as the 

 ordinates of a parabola : some continual- 

 ly decrease, so that their minimum state 

 is nothing, as the ordinates to the asymp- 

 totes of the hyperbola. Others increase 

 to a certain point, which is their maxi- 

 mum, and then decrease again ; as the 

 ordinates of a circle. Others admit of se- 

 veral maxima and minima ; as the ordi- 

 nates of the curve (fig. 5 ) a b c d e, &c. 

 where b and d are the maxima, and ace 

 are minima : hence it is easy to imagine 

 of other variable quantities, exhibited by 

 the ordinates of other kinds of curves. We 

 have, under the article FLUXIONS, given 

 some examples on the maxima and mini- 

 ma of quantities : we shall in this place 

 point out another mode of performing the 

 same thing, with uu example or two. The 

 rule is this : " Find two values of an or- 

 dinate expressed in terms of the abscissa: 

 put those two values equal to each other, 

 striking out the parts that are common to 

 both, and dividing- all the remaining 

 terms by the difference between the ab- 

 scissas, which will be a common factor in 

 them : then supposing the abscissas to be- 

 come equal, that the equal ordinates may 

 concur in the maximum or minimum, that 

 difference will vanish, as well as all the 

 terms of the equation that include it, and 

 therefore, striking those terms out of the 

 equation, the remaining terms will give 



the value of the abscissa corresponding to 

 the maximum." 



1. Suppose it were required to find the 

 greatest ordinate in a semicircle K M Q 

 Z. Let KZ=a .- K L the abscissa =x : L 

 M the ordinate = y : hence L Z =a x t 

 and by the nature of the circle KLxLZ= 

 L M 2 , that is a x o: 2 =:t/ z , 



Let the abscissa K Px x d, d being 

 equal to L P ; the ordinate P Q=LM = 

 y. K P X P Z = P QS or r + 'd X 

 a x d = a x x 2 2 d x-\-a d d 2 

 =z/ 1 = ax x- ; therefore 2 d x -{- 

 a d d-=0 : or a d = 2 d x -\-d- t or == 

 2 x -J- d, an equation derived from the 

 equality of the two ordinates : now, by 

 bringing the two equal ordinates toge- 

 ther, or making the two abscissas equal, 

 their difference, d t vanishes, and a=2 x, 



or x = - = K N, the value of the ab- 

 scissa K N, when N O is a maximum, that 

 is, the greatest ordinate bisects the dia- 

 meter. 



2. Let it be required to divide a given 

 line into two such parts, that the one 

 drawn into the square of the other may be 

 the greatest possible. Let the given line 

 be a ; one part x, of course the other part 

 a x and therefore by the terms of the 

 question x- x a: = ax 2 x"> is the 

 product of one part by the square of the 

 other. For the sake of comparison, let 

 one part be x-\-d, then theotherpart will 

 be a x d and a--f-Jl a X ti x d = a 

 X * X 3 3 dx* -f 2 a d - 3 d 1 x x -f- 

 a d* d3 (as before) a x 2 x3 . there- 

 fore, 3d x 3 - -\-T~a~tl 3 d- x x -|~ 

 a d z do, divided by d, gives 3 x 1 -j- 

 2 a 3 d x x -f- a d d-, and now strik- 

 ing out the terms that have d in them, 

 we get 3 OL * -f 2 a x = 0, and 3 x = 



2 a, and x = - , that is, the given line 



must be divided into two parts, in the ra- 

 tio of 3 to 2. 



MAXIM US (TYRIUS), in biography, a 

 celebrated philosopher and elegant writer 

 in the second century, was a native of 

 Tyre, in Phoenicia, whence he took his 

 name. Suiclas says, that he lived under 

 the Emperor Commodus, while Eusebius 

 and Syncellus place him under Antoninus 

 Pius. If we suppose that he flourished 

 under Antoninus, and lived to the time of 

 the first mentioned Emperor, the accounts 

 of those chronologers may be reconciled. 

 According to some writers, he can|e to 

 Rome in the year 146 ; where the Emperor 



