predictions by which the Son had given utterance to it fail? Impossible. Yet immediate appalling death even now threatens the struggling Saviour! His innocent soul, shrinking back from this unexpected cup, deprecatingly implores its removal! But what if the will of the Father should be to permit this dreaded catastrophe? Must a total "failure" ensue? Must a fearful final "break-down" result? O, here comes in the faith of Abraham, of whom Jesus was at once a son and an antitype. Could not God as easily raise up Jesus from this death to fulfill the divine purpose and pledge of crucifixion, as Abraham had once believed him able to raise up Isaac from the dead to be a progenitor of Christ? It was not difficult for the faith of Jesus to grasp this idea. The crucifixion was a fixed unalterable fact in his mind; and though the marshaled hosts of hell might even overcome his helpless humanity ere it should reach the cross, yet God was faithful, and would certainly quicken that humanity to accomplish the appointed sacrifice of Calvary.

Was it not this consideration which prompted the resignation clause of Christ's prayer? He certainly knew that the Father's will involved his final death by crucifixion, but conceived that it might also involve surprising antecedents, even death in the garden and a resurrection from its blood-stained soil, preparatory to death on Calvary, and a resurrection from the new tomb of Joseph.

To this solution of the question the objection may be urged, that the Scriptures contain no intimation of any such faith as we have ascribed to Christ. To which we answer, that there is no intimation in the Old Testament of the nature, objects, and extent of Abraham's faith. "The Gospels and the epistles," .says Kitto, "clearly tell us wherein lay that faith of Abraham which was counted to him for righteousness;" and then adds: "It must be admitted that on the surface of the narrative the expectation and hopes of Abraham are temporal, and the promises also. It is refreshing thus, by the aid of the later Scriptures, to penetrate to their inner meaning and find that they were not such." So when St. Paul in the eleventh of Hebrews says, "By faith Abraham, when he was tried, offered up Isaac accounting that God was able to raise him up even from the dead," he reveals a phase of the patriarch's faith which it

...

is not possible for us to gather from the twenty-second of Genesis, or any other portion of the Old Testament. It was reserved to the supplemental revelations of the New Testament to disclose to us the full extent of his trust in God. And as the Holy Ghost was thus not pleased to state all the facts connected with his trial in the Hebrew Scriptures, so he may have judged it fit to withhold from the writers of the Christian Scriptures similar facts connected with the agony of Jesus. But were it the will of God that an additional written revelation should be made to men, we might hear another Paul speaking thus of him who took not on him the nature of angels, but the seed of Abraham: "By faith Jesus, when his soul was exceeding sorrowful even unto death, resigned himself to the will of his Father, accounting that he was able to raise him up even from the dead to make atonement on the cross.'

Other objections may occur to the reader, but we need not now pause to anticipate them. Enough has been written, if not to settle the question, at least, we hope, to awaken inquiry on the subject, and to call forth other and abler discussions of it.

ART. IV. ARITHMETIC.

A Higher Arithmetic, Embracing the Science of Numbers and the Art of their Application. By A. SCHUYLER, A. M., Professor of Mathematics in Baldwin University. New York: Shel

don & Co.

ARITHMETIC is the science of numbers and the art of their application. The reciprocal relation of science and art is, to all who cultivate either, a subject profitable for reflection. Science requires a basis of facts, and this basis is furnished by the discoveries made to meet the necessities of art. Thus the art of speaking preceded grammar, since the facts of speech form the basis of the science of language; the art of thinking preceded logic; the art of navigating, the science of navigation; and the art of war, military science. In its primitive form, art is necessarily crude and imperfect; but the facts empirically discovered to subserve the purposes of art are classi

fied in the light of the principles of reason. To the facts thus classified are applied the processes of induction and deduction, by which are developed other facts and principles which are also classified, thus constituting a system of truth called science. Science now, in turn, reacts on art, rendering its processes more rational and perfect. Again, art furnishes science with its peculiar language, consisting of its technical terms, its nomenclature, and its system of notation, thus contributing greatly to its flexibility and efficiency. Art without the aid of science is crude and imperfect, and science without the aid of art is chimerical and futile. As art in its incipient state precedes science, so the art of calculation preceded the science of numbers. It may not be uninteresting or unprofitable briefly to trace the progress of Arithmetic from its crude beginning to its present advanced state of perfection.

The art of calculation must, at least in its rudimental form, have been coeval with the first stages of civilization. The origin of Arithmetic is not, therefore, to be referred to any one nation in particular, to the exclusion of other nations; for since it is indispensable in commercial transactions, and even in the ordinary business of life, it must have been understood to some extent, however imperfectly, by the first generations of civilized man.

The progress of the development of any science depends, to a great extent, on the facilities at the command of those who cultivate it; and among these facilities, by no means the least, is the perfection of the language or system of notation by which the elementary facts of the science together with their relations can be concisely expressed; and in no science is this more strikingly exhibited than in Arithmetic. We have historical evidence that Arithmetic was carefully cultivated by the Greeks. Both Thales and Pythagoras, who traveled to the East in search of light, cultivated Arithmetic with great success; but the labors of Pythagoras, valuable as they were, degenerated into the chimerical scheme of attempting to account for the facts of the universe by referring them to the properties of numbers. According to the Platonists, "Arithmetic should be studied, not with gross and vulgar views, but in such a manner as might enable us to attain to the contemplation of numbers; not for the purpose of dealing with mer

chants and tavern-keepers, but for the improvement of the mind, considering it as the path which leads to the knowledge of truth and reality."

The Greeks represented numbers by the letters of their alphabet. In this respect they seem to have imitated the Hebrews; for, as they had no letter corresponding to the sixth letter of the Hebrew alphabet, they represented the number six, not by the sixth letter of their alphabet, but by another character. The following was the Greek notation:

The tens by

2.

10, 20, 30, 40, 50, 60, 70, 80, 90.
L'. K'. 2. μ. v. §'. Oʻ. π'. 2.

The hundreds by 100, 200, 800, 400, 500, 600, 700, 800, 900.

ρ. o'. T'. v.

. x'.

ป.

The thousands by 1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000,

a. B. 1. 8. ε.

Ten thousands by 10000, 20000, etc.

K.

9000. 5. 5. M. 10.

The letter M was also used to denote a myriad, or ten thousand; and M placed under other expressions for numbers multiplied by ten thousand; thus φκδ' denotes 5240000.

M

By a proper combination of the letters, and by the extension of the scheme of notation exhibited above, the Greeks could express any number whatever.

Though this system of notation is vastly inferior to the one now in vogue, yet the Greeks displayed great ingenuity in its use, and made considerable progress in the science of numbers. In this respect they far excelled the Romans. In fact, the system of notation adopted by the Greeks was much superior, for the purpose of calculation, to that adopted by the Romans, as it is more analogous to the present system. To ascertain how ill-adapted the Roman notation is to the purposes of calculation, let an attempt be made to multiply together, for example, the numbers MDCCLXXIV and DCCCXCVIII. No wonder the Romans thought the operation of calculating a drudgery fit only for slaves!

To obviate the difficulties in calculating resulting from their imperfect notation, the Romans resorted to calculating machines, one of which, the most common, was the abacus, or a board on which were placed calculi, or pebbles, by the various

arrangements of which the calculations were performed. The modern abacus, or numeral frame, used in primary schools to aid the mind of the young learner in comprehending the relations of numbers, may be considered the Roman abacus modified and brought to perfection by modern ingenuity.

For the system of notation now in vogue called the Arabic, which possesses so many and such decided advantages over all other systems, the Europeans are indebted to the Arabians, who in turn are indebted to the Hindoos. The characteristics of this system are the simplicity of its characters, the decimal scale, and the device of place.

In respect to the characters 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, let it suffice to say they possess, in a high degree, the characteristics of simplicity and beauty.

The decimal scale was undoubtedly suggested by the primitive mode of reckoning by the ten fingers and thumbs. The following conventional principles form the basis of the decimal system:

One, or a unit of the first order, is the primary basis of all numbers. Ten units of the first order equal one unit of the second order; ten units of the second order equal one unit of the third order; in general, ten units of any order equal one unit of the next higher order. Numbers are thus considered. as formed into collections or groups, according to the scale of ten. Eight, twelve, or any other number might have been adopted as the scale; hence the decimal scale is to be considered, not as essential to a system of Arithmetic, but as conventional. Some have thought that a duodecimal scale and twelve characters would have afforded additional facilities for calculation, since twelve has a greater number of aliquot parts than ten. Thus, of twelve, the aliquot parts are its halves, thirds, fourths, and sixths; while of ten, the aliquot parts are its halves and its fifths. If, however, we consider how large a place in calculating, and in the various mathematical tables is occupied by decimal fractions; and if also we consider that similar fractions formed according to the duodecimal scale would not possess superior advantages, the superiority of the duodecimal. scale, if still claimed, is reduced to an inconsiderable value. The decimal scale answers every practical purpose, and mathematicians generally are satisfied with it. But the crowning excellence of the Arabic system is the device of place.