SQU 
Some properties of squares are as follow : 
1. Of the 
Natural series of squares 1^, 3% 4\ &c. 
which are equal to . . . .*.1 , 4 , 9 , 16 , &c. 
The mean proportional m n between any 
two of tliese squares m? and rf, is equal to 
the less square plus, its rootmultiplied by the 
difference of the roots ; or also equal to the 
greater square minus its root multiplied by 
the said difference of the roots. That is, 
mn dm = — d m ; 
where d^n — m is the difference of 
their roots. 
2. An arithmetical mean between any 
two squares and n^, exceeds their geo- 
metrical mean, by half the square of the 
difference of tlieir roots. 
That is -J- = mn-\- \d^. 
3. Of three equidistant squares in the 
series, the geometrical mean between the 
extremes, is less than the middle square by 
the square of their common distance in the 
series, or of the common difference of their 
roots. 
That is, mp = n^ — ; 
where m, n, p, are in arithmetical progres- 
sion, the common difference being d. 
4. The difference between the two ad- 
jacent squares and n^, is = Sm 
-4- 1 ; in like manner, 1, 
the difference between the next two adjacent 
squares and p^ ; and so on, for the next 
following squares. Hence the difference of 
these differences, or the second difference of 
the squares, is gn — 2m = 2 x n — m — 2 
only, because n — jn = 1 j tliat is, the se- 
cond differences of the squares are each the 
same constant number 2 ; therefore the 
first differences will be found by the conti- 
nual addition of the number 2 ; and then 
the squares themselves will be found by the 
continual addition of the first difference ; 
and thus the whole series of squares is 
constructed by addition only, as here 
below : 
2d Diff.....l 
2 
2 
2 
2 
2 
2 
&c. 
1st Diff. 
1 
o 
3 
7 
9 
11 
13 
&c. 
Squares 
1 
4 
9 
16 
^25 
36 
49 
Sec. 
5. Another curious property, also noted 
by the same author, is, that the sum of any 
number of the cubes of the natural series 
1, 2, 3, 4, &c. taken from the beginning, gl- 
ways makes a square number, and that the 
STA 
series of squares, so formed, have for their 
roots the numbers....!, 3, 6, 10, 15, 21, &c. 
the diffs. ofwhicharei,2, 3, 4, 5, 6, &c. 
viz. 
l’ = H, 
1*+2’=3S 
l’^-2* + 3^=6^ 
l3_j_g’_|_33_j_43_io2 . and in general 
1^ 4- 2 ^ -f 3 ^ = (1 + 2 4- 3 + ^ 
= I where n is the number of the 
terms or cubes. 
Squaring the circle, is the making or 
finding a square whose area shall be equal to 
the area of a given circle. The best ma- 
thematicians have not yet been able to re- 
solve this problem accurately, and perhaps 
never will. But they can easily come to 
any proposed degree of approximation 
whatever ; for instance, so near as not to 
err so much in the area, as a grain of sand 
would cover, in a circle whose diameter is 
equal to that of the orbit of Saturn. The 
following proportion is near enough the 
truth for any real use, viz, as 1 is to 
.88622692, so is the diameter of any circle, 
to the side of the square of an equal area. 
Therefore, if the diameter of the circle be 
called d, and the side of the equal square s j 
then is s = .88622692d = || d nearly. 
Square root, a number considered as 
the root of a second power or square num- 
ber ; or a number, by whose multiplication 
into itself, a square number is generated. 
Square buttle, or Battalion of Mem, is 
one that hath an equal number of men in 
rank and file. 
Square, hollow, in the military art, is a 
body of foot drawn up with an empty space 
in the middle for the colours, drums, and 
baggage ; faced and covered by the pikes 
every way, to keep off horse. 
Square, an instrument consisting of two 
rulers, or branches, fastened perpendicu- 
larly at one end of their extremes, so as to 
form a right angle ; it is of great use in the 
description and mensuration of right angles, 
and laying down perpendiculars. 
Square, in naval affairs, is a term pecu- 
liarly appropriated to the yards and their 
sails, either implying that they are at right 
angles with the mast or keel, or that they 
are of greater extent than usual. 
SQUIRREL. See SciuRus. 
STACHYS, in botany, a genus of the 
Didynamia Gymnospei mia class and order. 
Natural order of Verticillatse, or Labiatas. 
Essential character : corolla upper-lip arch- 
