called Mathesis, THE DISCIPLINE, because of its incomparable superiority to other studies in evidence and certainty, and, therefore, its singular adaptation to discipline the mind. And this, notwithstanding these mysteries, (for are they not such ?) is the science, says the eloquent and profound Dr. Barrow," which "effectually exercises, not vainly deludes, nor vexa"tiously torments, studious minds with obscure sub"tleties, perplexed difficulties, or contentious disquisitions; which overcomes without opposition,

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triumphs without pomp, compels without force, and "rules absolutely without any loss of liberty; which "does not privately overreach a weak faith, but openly "assaults an armed reason, obtains a total victory, "and puts on inevitable chains." How does it happen, now, that when the investigation is bent towards objects which cannot be comprehended, the mind arrives at that in which it acquiesces as certainty, and rests satisfied? It is not, manifestly, because we have a distinct perception of the nature of the objects of the inquiry (for that is precluded by the supposition, and, indeed, by the preceding statement); but because we have such a distinct perception of the relation those objects bear one toward another, and can assign positively, without danger of error, the exact relation, as to identity or diversity, of the quantities before us, at every step of the process. Mathematics is not the science which enables us to ascertain the nature of things in themselves;-for that, alas! is not a science which can be learned in our present imperfect condition, where we see through a glass darkly ;"-but the

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science of quantity as measurable, that is, as comparable and it is obvious, that we can compare quantities satisfactorily in some respects, while we know nothing of them in others. Thus we can demonstrate, that any two sides of a plane triangle are, together, greater than the third, by showing that angles, of whose absolute magnitude we know nothing, are one greater than the other; and then inferring the truth of the proposition, from the previously demonstrated proposition, that the greater angle in a triangle is subtended by the greater side. So again, when we affirm that between any two consecutive terms of the natural series of whole numbers, there may be interposed an indefinite number of magnitudes which are not fractional, the reason at first revolts as if we proposed an absurdity; for it seems repugnant to the first principles of common sense that between 99 and 100, for example, it should be possible to interpose a multitude of numbers, none of which can be correctly represented by either 99 plus a fraction, or 100 minus a fraction. Yet, far from involving absurdity, the proposition is so strictly true, that we cannot refute it without rasing to its foundation all mathematical science. For, it is demonstrable that the square roots of 9802, 9803, 9804, 9805, &c. to 10000, are each, in succession, greater than the former, and the first of them greater than 99. In like manner we can prove that the cube roots of 860300, 860301, 860302, &c. to 1000000, are each in succession greater than the former; that the cube root of 860300, the smallest of them, while it exceeds 99, is less than the square root of 9802. In

like manner we can assign separate series of biquadrate and sursolid roots still more numerous than the square and cube roots, all of which shall be demonstrably unequal to each other, shall be interposed in point of numerical value between 99 and 100, and yet shall, none of them, be correctly expressible either by the sum of 99 and a fraction, or by the difference of 100 and a fraction. Here, then, reason must bend, put on the "inevitable chains," and feel itself constrained, not merely to acknowledge the existence of those incommensurables which are neither fractional nor integral numbers, but also that while they are unsusceptible of precise appreciation, they admit of as accurate comparison as any other mathematical quantities. No mathematician can tell the precise value of 2 or 5; every one can tell the precise value of 4 or 9: no one, nothwithstanding, will hesitate longer to declare that 2, than to declare that 9 exceeds that 3 is greater than 2.

5 exceeds

4, that is,

Once more, we cannot possibly know ALL the terms

because such knowledge implies a contradiction: neither can we know all the terms of the infinite series

yet we can show that these series are equal. For we can demonstrate that the first series is an expanded

function, standing with the quantity

1

a + c

in the re

lation of equality; we can likewise demonstrate, that the second series bears the relation of equality with

while a and c stand as general representatives of any quantities; yet those fractions must necessarily be equal, and thence we infer the like equality between the sums of the two infinite series. In a similar manner we can have no clear conception of the nature of the quantities a, b, &c.; yet we are as

30 = 50: since we can demonstrate that equality subsists in the former expression as completely as we can in the latter, both being referable to an intuitive truth. Every mathematician can demonstrate strictly that the conclusions he obtains by means of these quantities, though he cannot comprehend them in themselves, must necessarily be true: he therefore acts wisely when he uses them, since they facilitate his inquiries; and, knowing that their relations are real, he is satisfied, because it is only in those relations that he is interested.

To you, my friend, who are so conversant with mathematical subjects, this enumeration of particulars would be perfectly unnecessary, were it not in order to recommend that similar principles to those which I have here traced be adopted, when religious topics are under investigation. We cannot comprehend the

nature of an infinite series, so far as that nature depends upon an acquaintance with each term; but we know the relation which subsists between it and the radix from which it is expanded: we cannot comprehend the nature of the impossible quantities ✔a, ✔ - by b, &c.; but we know their relation to one another, and to other algebraic quantities. In like manner (though I should scarcely presume to state such a comparison, but for the important practical inference which it furnishes), we cannot, with our limited faculties, comprehend the infinite perfections of the Supreme Being, or reconcile his different attributes, as as to see distinctly how "mercy and peace are met together, righteousness and truth have embraced each other; or how the Majestic Governor of the universe can be every where present, yet not exclude other beings; but we know, or at least may know (if we do not despise and reject the information graciously vouchsafed to us by the God of truth), his relation to us, as our Father, our Guide, and our Judge.—We cannot comprehend the nature of the Messiah, as revealed to us in his twofold character of "the Son of God," and the "Man "Christ Jesus;" but we know the relation in which he stands to us as the Mediator of the New Covenant, and as he "who was wounded for our transgressions, "who was bruised for our iniquities, and by whose

ઃઃ

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stripes we are healed: "—Again, we cannot comprehend, perhaps, why the introduction of moral evil should be permitted by him "who hateth iniquity;" but we know, in relation to ourselves, that he hath provided a way for our escape from the punishment due