98 Mr. Woodhouse on the Truth of Conclusions 
1 — x s/ — 1 — — , &c. ^ + 3 ) ^ ~ 1 ai-i abridged symbol 
for 1 + (*£ -f y) s/ — 1 — ^ r t + — &e. and there can be no 
ambiguity in what these symbols are meant to represent ; since 
we have only in the demonstrated form 1 x + -j— + &c. to 
substitute x s/ — 1, — x s/ — 1, or (x -f y) </ — 1, for x. 
The use made of these abridged symbols is, to express, in an 
algebraic form, certain lines belonging to the circle, as sines, 
cosines, &c. for, since 
e x>/ — 1 is an abridged symbol for 1 -f xV — 1 — ~ — "TTT’ & c * 
and e~ xV ~ 1 for 1 — x</ — 1 — — 4 - See . 
1.2 1 1.2.3’ 
+ e 
- x v~l 
xV~ 
2 V — 
— — - is a symbol for x 
is a symbol for 1 — i -- Sec. 
J 1 . 2 • 1 . 2 . 3 . 4 
X* 
1.2-3 • X.2. 3. 4. 5 
&C. 
but 1 — -{- &c. and x — \- Sec. represent the cosi 
and sine of an arc x 
+ e~ 
-, and*- 
cosme 
-*virr 
2 a/ — I 
in consequence of the assumptions made, properly represent 
the sine and cosine of an arc x. 
* The usual method of deducing these expressions, is by a fluxionary process. I 
have preferred an algebraical ohe, for the sake of perspicuity. In an algebraical in- 
vestigation, every step may be closely examined, and we can easily retrace to the orb 
ginal notions from which it commenced. In fluxions, the significancy of the expres- 
sions, and the nature and manner of their derivation, demand much time and attention, 
to be properly understood. If, however, the fluxionary process be examined, its object 
will appear to be, to find out a method of abridgedly representing the sine, &c;-of an 
arc, employing for that purpose a form demonstrated for real quantities. In this 
fluxionary process, it is quite unnecessary to mention either the hyperbola or. logarithms. 
