D d / are like Triangles ( or differing Infinitely little from 
loch 4 therefore CafBb J: A a : : <r>d : D d- and coni' 
poundjng B b 4 A a : A a : : / d 4 D d : D-d. Ake* 
natino- B b 4 A a: Jtd D d . : A a : D d. But B b + 
A a ==•■ v d 4 D d -( as is fliewn above ) therefore A a = 
D d. A a is the fluxional Particle of the Curve F A, and 
Dd is the fluxional Particle of the Line HD: Thefe 
Fluxions or Augments, being equal, and their flowing 
quantities beginning together, are themfelves therefore 
equal, viz- F A = B D. 
Let F G = x. G A ( — H B J y. AD = t. BD- 
S. So is the Curve F A = H D = y + S: that is, the 
Curve from the Vertex to any given point therein , is equal to 
the Sum of its Ordinate, and .Subtangm to the fame point 
ivhich is its fecond Property* 
HI. The next Property ( and whereupon I call it the 
Byperholic guadratrix ) is this, In Fig. 2. let F A E be a 
part of the Curve, ( &c . as before.) F I K II is a Squ re 
upon theline F Hr A I L is an Equilacer Hvper. -1 3 
Vertex is I, its Afymptotes H O, H fl. its Ax H > p . Fr no 
a given point L in the Hyperbola ( below its V< rte ) 
draw L A parallel to the Afymptote B. H inurfeC g 
the Diagonal I Hin M, F H in G, and touching th 
dratrix in A. I fay, that the Hyperbolic Are i I ; s 
equal to a Rectangle, whofe fides- are th & Ordinal <j A, 
and twice F B, the Ax to the Quadratrix, that is, V film.. 
I L M = 2 F H*GA. 
Let F H = a, FG = x, G A — ■ y. Beeaufe of the- 
Hyperbola G L X G H ( L S ) =■ F H q. therethr: Q L 
’ and 
Jjf M = • ~Y X 
the flexion of tlr 1 fc.M 
2 a x — xx 
a x 
