MAX 
Academy of Berlin, from the year i'i'46 to 
1756. 
MAURITIA, in botany, belonging to 
the App. Palniae, and natural order of 
Palms. Essential character : male in an 
oblong sessile anient ; calyx one-leafed, 
cup-shaped, entire ; corolla one-petalled, 
with a short tube, and a three-parted bor- 
der ; filaments six. There is but one spe- 
cies, rir. M, flexuosa, a native of the Woods 
of Surinam. 
MAXILLA, the jaws, or those parts of an 
animal in which the teeth are set. 
MAXIM, an established proposition or 
principle, in which serise it denotes much 
the same with axiom. See Axiom. 
Maxims are a kind of propositions, which, 
have passed for principles of science, and 
which, being selfevident, 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; lienee it stands opposed to the 
minimum, which is the least possible quantity 
in any case. Thus in the expression 
b X, where a and b are constant, and x vari- 
able, the value of the expression will increase 
as 6 a; or x diminishes, and it will be great- 
est, or a maximum, when x is least, or s= 0. 
The expression a} — ~ increases as - dimi- 
X X 
nishes, that is as * increases, and it will be 
a maximum when x is infinite. If along the 
diameter, KZ (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 great- 
est, and from thence it decreases till it va- 
nishes at Z. Some quantities continually 
increase, and have no maximum, unless 
what is infinite, as the ordinates of a para- 
bola : some continually decrease, so that their 
minimum state is nothing, as the ordinates 
to the asymptotes of the hyperbola. Others 
increase to a certain point, which is their 
maximum, and then decrease again ; as 
the ordinates of a circle. Others admit 
of several maxima and minima; as the 
ordinates of the curve (fig. b.) ab c d e, &c. 
where b and d are the maxima, and ace are 
minima : hence it is easy to imagine of 
otlier variable quantities, exhibited by the 
ordinates of other kinds of curves. We 
have, under the article Fluxions, given 
some examples on the maxima and minima 
ot quantities, we sliallin this place point out 
another mode of performing the same tiling, 
with an example or two. The rule is this : 
VOL, IV, 
MAX 
“ Bind two values of an ordinate expressed 
in terms of the abscissa ; put those two va- 
lues equal to each other, striking out the 
parts that are common to both, and dividing 
all the remaining terms by the difference 
betvi'eeli the abscissas, which w'ill be a 
common factor in them : then isnpposing 
the abscissas to become equal; that the 
equal ordinates may concur in the maxi- 
mum or minimum, that dilference will va- 
nish, as well as all the terms of the equa- 
tion that include it, and therefore striking 
those terms out of the equation, the re- 
maining 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 K Z « ; K L the abscissa == a- ; L M 
the ordinate y : hence L Z = a — x, and 
by the nature of the circle K L x L Z zs: 
L M“, that is a X — x^ tez if. 
Let the abscissa K P x X d) d being 
equal to L P ; the ordinate P Q — ■ L M = 
y. K P X P Z = P Q\ or. .x- -f d X 
a — X — d=i ax — f d x ad — d‘ 
z=zy^~ a X — x^ ; therefore — 2 d a; 
ad — d^ =: 0 : or a d ^ d x d}, or a rs 
2 X -j- d, an equation derived from the equa- 
lity of the two ordinates; now, by bringing 
the two equal ordinates together, or making 
the two abscissas equal, their difference, d, 
vanishes, and a 2 a;, or a; = ? = K N, the 
’ ’2 
value of the abscissa K N, when N O is a 
maximum, that is, the greatest ordinate bi- 
sects the diameter. 
2. Let it be required to divide a given 
line jnfo 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 a — X = a — .a;’ is the product of 
one part by the square of the other. For the 
sake of comparison, let one part be a;-{~d, 
then the other part will be a — x — d and 
a- -1- ol ^ X a — X — d^a x^ — x^ — Sdx^ 
-j- 2 a d — 3 X X ^ ad^ — (as be- 
fore) a x^ x’’: therefore, — 3d x^ -j- 
a ad — 3 d^ X . 1 ’ -j- n d^ — d 3, divided by 
d, gives — 3 -j- 2 a — 3d X a; -[- a d — d^, 
and now striking out the terms that have d 
in them, we get — 3 x- Sax = 0, and 
2 
3 a: = 2 a, and x — a ; that is, the given 
line must be divided into two parts, in the 
ratio of 3 to 2. 
X ' 
-J 
