- es hence it is evident that the greater the value 
of X, the less will be the value of v; but the perimeter 
_ _ & being given, the area-X is the, ble when 
"the polygon is regular, (7.),, therefore the of 
CE tee Sete 
apg mag aaa na 
same area | 
‘ Prtor. XII. ° Pireon. 
Of regular poly, having the same area; that 
which has the Tees ronal of sides has the least pe- 
rimeter. 3 a | ~ & 
_ Let v and z be the peri of two regular poly- 
__ gons, having the same area A ; also let Y and Z be the 
_ areas of two polygons, similar to them which have the 
. ema peunnten ee then, because of the similar polygons, 
i Bae Ot eb? san: 2 Vy : 
Ls berate: Ze A, 
») thence, ex eq. v*: 2*:: Z: ¥. ; 
Now, if of the two. polygons Y and Z, Y be that which 
has the greater number of sides, then Y will be greater 
Z (9.),.and consequently ZY; therefore v? will 
be x*, and v less than z; that is, the perime- 
ter of the polygon having the greater number of sides, 
is less than the perimeter of the other polygon. 
=< °° Prop, XIII. ~Tueror. 
The perimeter of a circle is less than that of any po- 
lygom having anvequal area, © ; 
te < = =) he sitod 
. This proposition may be’ proved 
pes teeter 
is the limit er s can i- 
bly be described about it; an slot while Pap 
scp, wan differ from the area of the polygon by 
any assi e tity, its perimeter will be 
See 
Cor. A circle contains a given area with the least 
possible perimeter, é 
SECT. II. 
exactly in the same 
sidering that a circle 
Tue ConstTRucTION oF GEoMETRICAL ProBLEMS, BY 
DESCRIBING CIRCLES ONLY. 
A etrical problem: is considered as resolved, 
9- when it is shewn to be identical with some other known 
problem, ‘or to be a combination. of several, the mode 
of resolving each of which is known. The decompo- 
sition-of a problem. into others more simple, leads to 
the que , which blems are the most simple ? 
wr Pesndbiganeias exoniette dee wick con 
; ; eters assumed, as the most el - 
cont oon ‘ ed, as the most elemen 
1. To — a straight line from any one point toany 
‘2. To produce a tefminated straight I 
length in eraghttine ~ ‘straight line to any 
“4 o describe a circle from an , i 
: . chat rir | ay centre at any dis« 
_, They did not propose to resolve these, but took for 
granted that their resolution was known, and as obvi- 
open te tuthof tiaies 
GEOMETRY. 
237 
However narrow a foundation these three self-evi- Appéndix. 
dent problems, or postulates as they are called, may 
ai to afford, when compared with the vast fabric 
0 i doce attempts have been made to render it 
still narrower. Tartalea proposed to Cardan, to con- 
struct all the problems in Euclid by one and the same 
opening of the compasses, admitting, however; the use 
of ‘a rule; and Benedictus composed a’ work on. this 
problem. Schooten, instead of the postulate; “ that a 
circle may be described from any centre at any distance 
from that centre,” substituted this, that from a given 
int in an indefinite straight line,’a straight line may 
cut off equal to a given terminated straight line ;” 
this change, he shewed elegantly how all the pro« 
blems in elementary geometry om be constructed, 
without employing the circle farther than to cut off 
from a linea part of a given length; and thus in a 
eee the problems were constructed by ‘straight 
ines only. See Schooten, Exercit.’ Math. lib. ii. 
At a later period, an Italian mathematician, Mas- Mascheroni. 
cheroni, imposed ‘on himself the task of resolving all Died 1801. 
plane problems whatever, by the circle alone ; his sue- 
cess was complete, and the result of his labours'is given 
in his Geometrie du Compas, the Geometry of the Com- 
passes. It must be observed, that it is only in the 
construction of the problem that the straight line® is 
dispensed with; for, in the demonstration, straight 
lines must be supposed drawn, and their properties in- 
troduced, in order to apply the common elements of 
geometry. het: 
It is an anecdote not altogether without interest in the 
history of geometry, that the celebrated Bonaparte, late 
Emperor of the French, studied the geometry of the 
compasses under Mascheroni ; he even condescended to 
Peps to the French mathematicians one of its pro- 
lems, namely, to divide the circumference of a circle 
into four equal parts, without employing straight lines. 
We shall now give some speciniens ‘of this mode of 
constructing problems. And it is to be observed, that 
the ositions referred to in the article Geometry, 
are all in the first Part. 
Prop. I. 
To determine a distance in the direction of a straight 
line passing through two given points A, B that shall 
be any srateipie of the given distance A B. 
Pros. 
First, To double the distance AB. On B asa cen- Fig. 181, 
tre, with BA as a radius, describe a portion ACDE of 
a circle, not less than its half.. On A as a centre, with 
the same distance, describe an arc, to cut the circle in 
C. In like mamner determine’ the points, D and E in 
the circumference, so that the distances from.C to D,, 
and from D to E, may be equal to the distance from A 
to C, or from Ato B. Then, because the chords of 
the ares AC, CD; DE are each equal to the radius AB ; 
the arc ACDE will errant a semicircle (2. .5.), and 
the points A, E will be the extremities of a diameter ; 
therefore the points A, B, E will lie in a straight line; 
and the distance AE will be double AB. 
2d, To find the triple of the distance AB. Take BF 
the double of BE, and AF shall be the triple. of AB ; 
and proceeding in this way, any multiple whatever of 
AB may be found, 
Prop, II. 
To divide a given distance AB into any proposed Fig. 182. 
number of equal parts. i 
1 
Pros. 
—_—y= 
