ness in the phrase mental experiments, which the Doctor uses, and which might have been avoided by a different mode of stating his argument or view; and which seems to be this, that the fundamental notions or conceptions from which mathematical reasoning starts, and to which it appeals, are as much the result of experience, and rest as much upon the evidence of the senses, and the natural meaning of our own words, in connexion with that evidence, as the fundamentals of any physical science whatsoever; and he instances particularly the axioms, as they are called, that "two straight lines cannot enclose a space," and "the whole is greater than its part.'

In a review of a treatise of Leslie's, on mathematics, attributed to Professor Playfair, in the twentieth volume of the Edinburgh Review, there are some remarks upon this work of Dr. Beddoes, which, coming from Professor Playfair, are entitled to particular consideration. Playfair sug

gests that Beddoes was no great mathematician. But with submission, this is no

answer to Beddoes' argument, and rather too near an approach to the common tactics of controversial writing, in which the reader's attention is diverted from the question, and the pursuit of truth, by some insinuation against the character or abilities of an adversary. Playfair tells us that "geometrical reasoning is a process purely intellectual, and resting ultimately on truths which the mind intuitively perceives." Are we, then, to rest here without going further,—without venturing to ask what are "truths intuitively perceived"? In what sense this is true, the present observations are meant to illustrate, and, if I am not very much deceived, will sufficiently, or in a great measure, help the reader to understand. Meantime I beg to call his attention to a remarkable and just sentence of Hartley's, in his invaluable and profound chapters on "Words, and the Ideas associated with them, and on Propositions and the Nature of Assent." "Rational assent to any proposition may be defined a readiness to affirm it to be true, proceeding from a close association of the ideas sug

gested by the proposition, with the idea or internal feeling belonging to the word truth, or of the terms of the proposition with the word truth;" and then follow some observations on geometrical and mathematical reasoning, which are as clear, beautiful, and unanswerable, as any observations upon abstract truths within the circle of human science and philosophy.

But, fourthly, whencesoever we get the notions or conceptions with which we are concerned in mathematical reasoning, I think it must be admitted, that habit, i. e. the constant recurrence of the same simple ideas of number and figure, and the constant association of the same terms with the same ideas, has much to do with that feeling of certainty and satisfaction, that readiness and confidence of assent, which we recognise in connexion with the processes of arithmetic, algebra, and geometry.

How much there is in habit may be easily and irresistibly shown. Thus we say that 2 and 3 make 5, and the three angles of a triangle are equal to two right angles;

and we feel the truth as we we pronounce the words. But if we take higher numbers, and more advanced propositions,-if we say that nine thousand six hundred and seventy'three (9673) times seventy-three thousand six hundred and nine (73,609) make 712,019,857, or upwards of seven hundred and twelve millions; or if we take some of the propositions relating to proportion in the fifth book, or go on to the more abstruse calculations in algebra, trigonometry, and fluxions, will our assent be so ready? Who will assert it? And why ?—because we are not in the habit of attending to high numbers and advanced propositions. Doubt, ignorance, and difficulty attach themselves to our terms. He who has just risen from calculations, or the study of mathematics, will feel a confidence in terms and propositions which others do not. A ready accountant casts up with a glance or two a long column of accounts; he perceives the relation of each item to the whole amount in a space of time that appears incredibly short to one wholly unaccustomed to such work.

Those who are in the habit of

38 FEWNESS OF TERMS AND PREMISES

estimating the number of persons in a crowded room or assembly, can tell by looking at the mass, with reference to the space occupied, how many may be present with much more correctness than another who should try for the first time to count the heads. So the bare statement of a proposition, and a glance at the diagram, will enable the quick mathematician to understand the whole demonstration, and to repeat the various steps of the process faithfully to another; while he who is slow at combining the ideas of figure, notwithstanding ever so careful reading of the proof, will be still at a loss to perceive its cogency; and will pass from the words to the figure, and the figure to the words, without being a whit the wiser, or having any distinct idea of what he is about, or where he is, present to the mind. The elaborate paper of Sir W. Hamilton, of Dublin, to the Royal Society*, appears a chaos of warring elements, a mere jumble of letters and figures to the tyro

* "On a General Method in Dynamics."-Phil. Trans. 1834, Pt. ii., p. 247.