its sign by passing, as mathematicians say, either through zero or through infinity. In the latter case there must be one intermediate value of the variable for which the variant is indifferently negative infinity and positive infinity. Mathematicians may shirk the difficulty, but they cannot make this common result of mathematical principles appear otherwise than contradictory to our common notions of space. The hypothesis that there is a Creator at once all powerful and all benevolent is surrounded, as it must seem to every candid investigator, with difficulties verging closely upon logical contradiction. The existence of the smallest amount of pain and evil would seem to show that He is either not perfectly benevolent, or not all-powerful. No one can have lived long without experiencing sorrowful events of which the significance is inexplicable. But if we cannot succeed in avoiding contradiction in our notions of elementary geometry, can we expect that the ultimate purposes of existence shall present themselves to us with perfect clearness? I can see nothing to forbid the notion that in a higher state of intelligence much that is now obscure may become clear. We perpetually find ourselves in the position of finite minds attempting infinite problems, and can we be sure that where we see contradiction, an infinite intelligence might not discover perfect logical harmony? From science, modestly pursued, with a due consciousness of the extreme finitude of our intellectual powers, there can arise only nobler and wider notions of the purpose of Creation. Our philosophy will be an affirmative one, not that false and negative one of Auguste Comte, which has usurped the name, and misrepresented the tendencies of a true positive philosophy. Our science will not deny the existence of things because they cannot be weighed and measured. It will rather lead us to believe that the wonders and subtleties of possible existence surpass all that our mental powers allow us clearly to perceive. The study of abstract logical and mathematical forms has seemed to convince me that even space itself is no requisite condition of conceivable existence. Everything, we are told by materialists, must be here or there, nearer or further, before or after. I deny this— and point to logical relations as my proof. There formerly seemed to me to be something highly mysterious in the denominators of the binomial expansion (vol. i. p. 216) which are reproduced in that strange natural constant e, or and in many results of mathematical analysis. I now perceive, as already partially explained (vol. i. pp. 40-42, 180, 181, 443, 444), that they arise out of the fact that the relations of space do not apply to the logical conditions which govern the numbers of combinations as contrasted to those of permutations. So far am I from accepting Kant's doctrine that space is a necessary form of thought, that I regard it as an accident, and an impediment to pure logical reasoning. Material existences must exist in space no doubt, but intellectual existences be neither in space nor out of space; they may have no relation to space at all, just as space itself has no relation to time. For all that I can see, then, there may be intellectual existences to which both time and space are nullities. may Now among the most unquestionable rules of Scientific Method is that first law that whatever phenomenon is, is. We must ignore no existence whatever; we may variously interpret or explain its meaning and origin, but if a phenomenon does exist it demands some kind of explanation. If then there is to be a competition for scientific recog nition, the world without us must yield to the undoubted existence of the spirit within. Our own hopes and wishes and determinations are the most undoubted phenomena. within the sphere of consciousness. If men do act, feel, and live as if they were not merely the brief products of a casual conjunction of atoms, but the instruments of a farreaching purpose, are we to record all other phenomena and pass over these? We investigate the instincts of the ant and the bee and the beaver, and discover that they are led by an inscrutable agency to work towards a distant purpose. Let us be faithful to our scientific method, and investigate also those instincts of the human mind, by which man is led to work as if the approval of a Higher Being were the aim of life. THE END. INDEX. ABACUS, the logical, i. 119; Briggs', Academie del Cimento, ii. 36, 42, 46. Accidental discovery, ii. 162. Airy, Sir George Biddel, ii. 171, 174, Alphabet, permutations of, i. 196, 203. Analogy, ii. 244, 283; in sign of equality, Apparent equality, i, 319; sequence of Approximation, principles of, ii. 92; to Aqueous vapour, ii. 126. 82. Arago, photometer, i. 335; rotating disc, Archimedes, de arena numero, i. 222; Aristotle, dictum, i. 26; overlooked Arithmetic, reasoning in, i. 188; of ap- Atoms, ii. 10; size of, i. 222; weights Augustin, on time, i. 359. Average, i. 417; divergence from, 213; B. Babbage, Charles, ii. 455; calculating Bacon, Francis, biliteral cipher, i. 220; Bacon, Roger, on rainbow, ii. 159, 247. Bain, on powers of mind, i. 4; induction, Baker, poem Universe,' ii. 276. Ballot-box, hypothesis of, i. 169, 275, Bernouilli, James, i. 302; combina- Bernouilli, numbers of, i. 143. of comets, 133; sun's parallax, 203; Bismuth, ii. 261. Bode's law, i. 165, 297. Boethius, i. 40, 83; on means, 418. Botany, ii, 335, 350, 353, 388; nomen- Bowen, on inference, i. 136; method of Boyle, law of, ii. 87, 91, 274; on hypo- Brewster, Sir David, refractive indices, British Museum, catalogue of, ii. 403, Buckle, ii. 321, 459- Bunsen, Robert, spectrum, i. 281; light, C. Calc-spar, ii. 325, 362. Canton, John, compression of water, i. 390. Capillary attraction, ii. 57. Carbon, conductibility of, ii. 53; che- Carbonic acid, ii. 334. Carnot, ii. 257. Catalogues, ii. 401. Cause, i. 253; probability of, 279. Centre of gravity, percussion, &c., i. Chalmers, on Collocations, ii. 434. Character, human, ii. 425. Chemistry, ii. 160, 205, 285, 347, 364, Circle, ratio of diameter and circumfer- Clairaut, ii. 314, 315; on gravity, 81. Clocks, astronomical, i. 330, 394; mu- Clouds, ii. 14, 15, 16. i. 35. Colours, iridescent, ii. 26; natural, 147; Combinations, i. 200; calculation of, 204. Comte, Auguste, i. 227, 245; ii. 171, Concrete number, i. 178. Conditions, i. 260; removal of usual, ii. |