TRANSACTIONS OF THE SECTIONS. 83 
On the Interpretation of certain Symbolic Formule and Extensions of 
Taylor’s Theorem. By the Rev. CHarves Graves, D.D., M.R.LA. 
On some Application of Quaternions to Cones of the Third Degree. 
By Sir W. R. Hamitton, LL.D. MRA. 
On the Icosian Calculus. By Sir W. R. Hamirton, LL.D., M.R.LA. 
The author stated that this calculus was entirely distinct from that of quaternions, 
and in it none of the roots concerned were imaginary, He then explained the lead- 
ing features of the new calculus, and exemplified its use by an amusing game, which 
he called the Icosian, and which he had been led to invent by it,—a lithograph of 
which he distributed through the Section, and examples of what the game proposed 
to be accomplished were lithographed in the margin, the solutions being shown to 
be exemplifications of the calculus. The figure was the projection on a plane of the 
regular pentagonal dodecahedron, and at each of the angles were holes for receiving 
the ivory pins with which the game was played. 
On Infinite Angles and on the Principle of Mean Values. 
By Mr. Commissioner HARGRAVE. 
On the Origin and Elimination of Euclid’s “ Reductio ad absurdum.” 
By Joun Pore Hennessy, of the Inner Temple. 
_ The author first pointed out the difference between direct and indirect demonstra- 
tion. The enunciation of every geometrical theorem is a conclusion. This conelu- 
sion may be proved in either of two ways: (1) either by the simple syllogistic method, 
or (2) by a combination of that method with the principle of opposition. Mr. Hen- 
nessy showed that every proposition which Euclid proves directly belongs to the first, 
and that every proposition proved indirectly belongs to the second class. The origin 
of the Reductio ad absurdum was thus resolved into the question, why the principle 
of logical opposition should be employed in some cases and not in others. He showed 
_ that the necessity for calling in the aid of logical opposition depended on two abnor- 
_ mal conditions: (1) when any of the premises of an affirmative proposition are nega- 
tive, and (2) when none of the premises of a negative proposition are negative. He 
_ then adverted to the number of indirect demonstrations which Euclid had left in the 
first six books of the ‘ Elements,’ and to the very small number of these which suc- 
_ ceeding geometers had altered. He concluded by submitting direct proofs of every 
_ proposition hitherto proved indirectly. 
On some General Propositions connected with the Theory of Attractions. 
. By the Rev. Professor JeLtett, M.A., M.R.I_A. 
The author showed that the attraction of a body whose particles act with a force 
varying inversely as any odd power of the distances, may be easily deduced from that of 
a body having the same form and quantity of matter, the law of attraction being the 
inverse first power of the distance. He showed further, that while a knowledge of 
the attraction of a body whose particles act according to the law of nature, that is 
_ to say, the inverse square of the distance, gives us no information as to any other 
_ law (at least in this method), the knowledge of the attraction for the inverse fourth 
_ power gives it for every higher inverse power, 
oy _ From these theorems he proved that there is no law of force capable of being 
_ represented by a series of inverse powers of the distance, except the law of nature, 
for which a body will attract as if its mass were concentrated at a fixed point. 
He showed also that there is no law of force capable of being represented bya 
finite series of inverse-powers of the distance, except the law of nature, for which a 
shell of any form will exercise no attraction on a point within it. If then the 
= hy * 
——. 
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