C 568 ) 
inquire, when AExBC — KExKC is a Mini- 
mum ? 
Suppofe the Point L to be fo taken upon K N, 
that KL may be to A L as KC is to BC. From 
the Centre A deferibe in the Plane A KE with the 
Radius AE } an Arc of a Circle E R meeting A L, 
produced, if necedary, in R 5 let E V be perpendi- 
cular to AR in V \ and KH be perpendicular to the 
fame in i/; then the T riangles LEV, L K H, LA K, 
being fimilar, we have LV: LE : : L H : L K : : 
LK: LA : : (by the Suppofition lad made) KC: BC . 
Hence, when E is between L and iY, we have 
LH+ LV ( = VH) : LK+LE (= KE) :: 
KC.BC-, and when E is between K and L> we 
have L H— LV(= VH ) : L K — LE (= KE) ; : 
KC:BCi that is, in both Cafes we have KEx 
KC — VHx BC 5 and confequently AExBC — 
KExKC==AExBC—VHxBC==AE~EEpr~H 
X B C = A R ' — VHx BC = AH+ V R x BC ; 
which, becaufe AH and B C do not vary, is evident- 
ly Lead when VR vanifhes, that is, when E is upon 
j L. Therefore CLBl is the Rhombus of the mod 
advantageous Form in refpeft of Frugality, when K L is 
toAL as K C is to BC. This is the fame Method by 
which we have elfewhere determined the Maxima 
and Minima , in the Refolution of feveral Problems 
that have ufually been treated in a more abdrufe Man- 
ner. See Treatife of Fluxions, Art. 5 72, &c. 
Now becaufe O K is bife&ed in A, KC 2 = 0 K 2 
=z 4. A K 2 i and AC 2 — 1 A K 2 , osBC=ziAC=z 
2 \/ ixAKj confequently KC.BC : : zAK: 2 -\/ 3 
x AK w 1 : V 3 ; and K L : A L ; ; {KC : BC) : : 
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