275. Now it is obvious that what Hume accounts for by means of this tendency to feign, even if the tendency did not presuppose conditions incompatible with his theory, is not mathematical science as it exists. It has even less appearance of being so than (to anticipate) has that which is accounted for by those propensities to feign, which he substitutes for the ideas of cause and substance, of being natural science as it exists. In the latter case, when the idea of necessary connexion has been disposed of, an impression of reflection can with some plausibility be made to do duty instead; but there is no impression of reflection in Hume’s sense of the word, no ‘propensity,’ that can be the subject of mathematical reasoning. He speaks, indeed, of our _supposing_ some imaginary standard--of our having ‘an obscure and implicit notion’--of perfect equality, but such language is only a way of saving appearances; for according to him, a ‘supposition’ or ‘notion’ which is neither impression nor idea, cannot be anything. A hasty reader, catching at the term ‘supposition,’ may find his statement plausible with all the plausibility of the modern doctrine, which accounts for the universality and exactness of mathematical truths as ‘hypothetical’--the doctrine that we suppose figures exactly corresponding to our definitions, though such do not really exist. With those who take this view, however, it is always understood that the definitions represent ideas, though not ideas to which real objects can be found exactly answering. Perhaps, if pressed about their distinction between idea and reality, they might find it hard consistently to maintain it, but it is by this practically that they keep their theory afloat. Hume can admit no such distinction. The real with him is the impression, and the idea the fainter impression. There can be no idea of a straight line, a curve, a circle, a right angle, a plane, other than the impression, other than the ‘appearance to the eye,’ and there are no appearances exactly answering to the mathematical definitions. If they do not _exactly_ answer, they might as well for the purposes of mathematical demonstration not answer at all. The Geometrician, having found that the angles at the base of _this_ isosceles triangle are equal to each other, at once takes the equality to be true of all isosceles triangles, as being exactly like the original one, and on the strength of this establishes many other propositions. But, according to Hume, no idea that we could have would be one of which the sides were precisely equal. The Fifth Proposition of Euclid then is not precisely true of the particular idea that we have before us when we follow the demonstration. Much less can it be true of the ideas, _i.e._ the several appearances of colour, indefinitely varying from this, which we have before us when we follow the other demonstrations in which the equality of the angles at the base of an isosceles is taken for granted.