## A new theory of parallels |

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accepted admit amount angular magnitude assert assuming Axiom base base-angle become believe Book certain Chord Circle common contains continuously Corollary course defect Definition describe difference direction doubt draw Edition equal equally inclined equidistant Euclid's Euclid's Axiom exceed exist Figure finite genuine given grant greater Hence Hypothesis imagine impossible Infinite Infinitesimal Infinity inscribed intersect isosceles Triangle join length less lies limit logical Magnitudes mean meet multiplied never not-less once opposite Pair of Lines Parallels pass perpendicular placed Point position possible possible region produced proof Prop Propositions provable proved question radius Ratio Reader Reason revolve revolving-Line right angles second order Sector seems Segment separational sides Space straight Line Strip suppose Tetragon Theorem Theory third transversal Triangle ABC Triangle whose angles true universally upper vary vertical angle whole width Zero

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Página xii - Pythagoras' theorem states that the square of the length of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the lengths of the other two sides.

Página 72 - Euclid and His Modern Rivals" containing a Notice of Henrici's Geometry, together with Selections from the Reviews. (London : Macmillan and Co., 1885.) WE noticed the original work at such length in these columns (NATURE, vol.

Página xii - Mathematical research, with all its wealth of hidden treasure, is all too apt to yield nothing to our research: for it is haunted by certain ignes fatui— delusive phantoms, that float before us, and seem so fair, and are all but in our grasp, so nearly that it never seems to need more than one step further, and the prize shall be ours! Alas for him who has been turned aside from real research by one of these spectres — who has found a music in its mocking laughter — and who wastes his life...

Página 71 - ... more than once done our little best in the same direction, when he recounts how, more than once, he, too, has " with clasped hands gazed after the retreating meteor, and murmured, ' Beautiful star, that art so near and yet so far.

Página xii - ... of the squares of the sides " is as dazzlingly beautiful now as it was in the day when Pythagoras first discovered it, and celebrated its advent, it is said, by sacrificing a hecatomb of oxen — a method of doing honor to Science that has always seemed to me slightly exaggerated and uncalled-for. One can imagine oneself, even...

Página xv - ... the three angles of a triangle are together equal to two right angles, although it is not known to all.

Página 51 - If a straight line cut two straight lines so as to make the two interior angles on the same side of it together less than two right angles, these straight lines...

Página xi - Papers (Cambridge, 1908), 2, p. 669. 302. It may well be doubted whether, in all the range of Science, there is any field so fascinating to the explorer — so rich in hidden treasures — so fruitful in delightful surprises — as that of Pure Mathematics. The charm lies chiefly ... in the absolute certainty of its results: for that is what, beyond all mental treasures, the human intellect craves for. Let us only be sure of something! More light, more light! 'Ei/ 8e <£aei Kal oXecrcrov " And if...

Página xi - And if our fate be death, give light and let us die!" This is the cry that, through all the ages, is going up from perplexed Humanity, and Science has little else to offer, that will really meet the demands of its votaries, than the conclusions of Pure Mathematics.

Página 67 - revolves through an equal angle, into the position of lying along AD,' for this would be to make AH fulfil two conditions at once. "If you say that the one condition involves the other, you are virtually asserting that the lines CF, AH are equally inclined to CD — and this in consequence of AH having been so drawn that these same lines are equally inclined to AE. "That is, you are asserting, 'A pair of lines which are equally inclined to a certain transversal, are so to any transversal.