78 
MR. DAVIES’S GEOMETRICAL INVESTIGATIONS 
added on account of its connexion as to form of enunciation with the third, and from 
its admitting of a very neat analysis and construction. 
The conclusion, I may, finally, add, at which I have arrived, is : — That when two 
centres of magnetic force of equal intensity and opposite direction are situated anywhere 
within the earth, there are always two, and never more than two, points on its surface at 
which the needle can tahe a direction perpendicular to the horizon . 
GEOMETRICAL LEMMAS. 
Lemma I. Local Theorem. — If from two given points lines he inflected to meet, and 
have a given ratio, the locus of their intersection is a given circle . 
This proposition was known to the Greek geometers, and is employed by Eutocius 
in his preface to the Conics of Apollonius. The analysis and synthesis of it are given 
by Simson, in two different ways, in his Restoration of the Plane Loci, lib. ii. prop. ii. 
The latter of these is also given by Professor Leslie in his Geometrical Analysis, 
book iii. prop. 13, and is that generally employed by geometrical writers. The fol- 
lowing one is, so far as I know, different from any that has been given : still, but for 
its better answering the purposes I have in view, I should not have inserted it here, 
as on no other account can it, perhaps, be entitled to such an appropriation. 
Let T and U (Plate X. fig. 1 .) be the given points, and T N, N U, a pair of corre- 
sponding lines in the given ratio. Bisect the interior and exterior angles TNU and 
UNK by the lines N C, N D meeting the line T U at C and D respectively. Then 
TC : CU : : TN : NU, 
and 
TD : DU : : TN : N U. 
Hence the sum, or the difference, of two lines and their ratio being given, the lines 
themselves are given, and hence the points C and D are given, and the line C D 
between them is given in magnitude and position. 
Again, since the interior and exterior angles at N, which are together equal to two 
right angles, are bisected by N C and ND, the angle CND is a right angle, and 
therefore also given in magnitude. And since CD is given, and the angle CND is 
a right angle, the locus of N is a circle on C D as a diameter. Hence the following 
Construction. — Divide the given line internally and externally in the given ratio in 
C D, and on C D describe a circle. This will, as is evident from the analysis, be the 
locus sought. 
Corollary. — The line TU is divided harmonically in C and D ; for from the two 
analogies above, we have 
TC : CU : : TD : DU. 
Scholium. — A similar division may be made, the points C and D lying on the other 
side of M, the middle of T U. This implies, however, an inversion of the antecedent 
and consequent of the terms of the ratio. 
