( 568 )

— P - - \spP + 4 ? h __

+

, . * ; srom which I derive the followlnc Riiie

i b 4 j, b 7 t. -

Divide hals the Parameter by the Horizontal Di(la>ice, and keep the .Quote,

P

viz- -; then siy, as Square os the Dißance given to the half Parameter ,

2. b

so half ParameterP p -+ 4fb

8

C Height 7

double < > to the Square of a Secant —

^.DescentS

,the Tangent anfwering to that Secant will be ^ — i,

4 bb 4 bb

or rr: so then the Sum and Difterence of the afore-found Quote and thisTangent , will be the Roots of the Equation , and the Tangent s of the Ele-vations iöught.

Note here, That in Defcents, is the Tangent exceed the Quote, as it doeswhen p h is more than b b, the Direflion of the Lower Elcvation will be be-low the Horizon, and if p b — bb, it must be Directed Horizontal, and the

Tangent of the upper Elcvation will be -—- : Note likewise, That if 4 bb

b

-+4 pb in Ascents, or 4 bb —.4 pb in Defcents, be equal to p p, there

is but One Elcvation that can Hit the Object, and its Tangent is- • and if

1 b

4 b b •+ 47? /» in Ascents, or 4 b b — 4 pb in Defcents, do exceed pp, theObject is withaut the Rpacb of a Prtjtct cäst with that Vclocity, and so thething impossible.

From this Equation 4 t b 4- 4/» b — p p, are determined Ehe utmost Li-mits of the IQeacb of any Project , and the Figure alügned, wherein are allthe Heights upon each Horizontal Distance beyond which it cannot paß ;

for by Reduction of that Equation, b will be found —.^p- -—

P

in Heights, and-. — L p in Defcents ; from whenee it follows, that all the

P

Points b are in the Curve of the Parabola, whose Focus is the Point fromwhence the Project is cast, and whose Latus IQcctum, or Parameter ad Axem•is ~ p. Likewise from the sime Equation may the least Parameter or Velo-city be tound capable to Reach the Object proposed ; for bb = % pp 4I p b

being reduced, i p will be = o/ b b -F iTls ^ ^ which is the

C — b in Defcents 3

■Horizontal Range at 45 0 . that wouldjust: Reach the Object, and the Elcvation

reejui-