signify spirit, separate, so the Christ was separate' from sinners, and though to us a child is born, a son is given.' He (wonder of wonders) is separate, incomprehensible, with the spirit of a child, and yet possessing all the attributes of God."

"The Desire of all Nations SHALL COME" (Haggai II, 7). This may be translated " the desirable things of all the nations," and so is remotely messianic, as the Christ would appear about the time of the completion of the Temple and give peace.'

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"THE STEM OF JESSE" (Isaiah IX, 1). Stump or stock of Jesse. This verse, running on through chapters xi and xII, introduces one of the most striking and distinctly messianic prophecies of the glorious reign of the Redeemer's kingdom to be found in the Bible."

The ten lines devoted to Shiloh appear to be extracted from the article on "SHILOH," in McClintock & Strong's "Cyclopædia " (Vol. IX, pp. 677-683), who devote thirteen columns to this word, and more than one column is in opposition to Prof. Delitzch's interpretations of the word Shiloh. The meagre replies to the other four question do not require a remark from us.

A two- or three-page article on any of these questions is solicited. for publication, with the writer's name; or a nom de plume, if the writer so desires for subscription.

CRITICISMS ON THE CHAPTER ON THE PROPERTY OF NUMBERS.

(VOL. X, P. 225.)

"Through the kindness of a friend, I have had the pleasure of reading the last four or five issues of your excellent periodical. Its perusal has been highly interesting to me in every respect, and I shall be happy to subscribe at once.

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In the September issue (1892), I noticed and carfully read an article entitled "A Chapter on the Property of Numbers," which is very interesting. Since it contains some errors, and a few points which require some explanation, I have taken the liberty of pointing them out. (Paragraph 60.) Denoting the second of the numbers by y, and the first by all the different relations mentioned are simple and easy transformations of the equation y+log=0, or yX101. They do not present striking peculiarities of the numbers, but are the natural results derived from certain easily understood operations performed on the latter equation. To any one not conversant with such operatins they probably will produce the impression of great singularity,

y

whereas to the mathematician they will appear as quite ordinary. (Parapraph 61.) The well-known relations of the base of thenatural system of logarithms and the modulus render the equations very simple and natural.

(Paragraph 62.) It should be stated that negative values are to be excluded, for, if they were not, the number of solutions would be infinite. Whether the result given is correct I have not had the time to ascertain.

(Paragraph 64.) There are some grave errors in some of the statements. If a polygon of 51 (=17X3) sides were constructible the trisection of an arc would not any longer be impossible. Nor is a polypon of 85 sides constructible, nor of 225 and 65535 sides, nor of any number save the first of those given in the last five numbers. Only those polygons are constructible, the number of whose sides satisfies the equation, 2" N, where n may be any interger, and N= 3, 5, 15, 17, 257, 65537,. 2m+1; 2m+1 being a prime

number.

(Paragraph 73.) The more accurate expression is 1(√/5—1) = x. Therefore, y=(√5+1), y={(√5+3), from which all the relations mentioned follow with far less trouble and labor than from the decimal fractions given.

(Paragraph 79.) The series given is not the only instance, since x + x2 + x3 + x2 + ....= y; y + y2 +3 +14 + .. show the same peculiarity.

1.

= x

(Paragraph 90.) The statement is not strictly true, since an √3' incommensurable quantity, exceeds its cube more than any other quantity. There is no decimal fraction which would express it accurately. The three decimals given express but a part of the required result." J. S., A. M.

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These chapters were designed to answer hundreds of questions that accumulated in our sanctum drawer, and many anticipated similar ones. Thousands of arithmetical scholars do not know what a logarithm is, but to a mathematician they are as simple as an abacus. However, it is a little more novel to the eye to observe the arithmetical results, than simply to state formulæ as in Paragraphs 60 and 73. 'Numbers are divine," thought Plato; and "Uneven numbers please the Gods," says Virgil; the mathematician realizes these thoughts in the higher mathematics, and all who are engaged in original research The errata appeared in December No., p. 334. The statements of Paragraph 64 are from Barlow's "Theory of Numbers," p. 505. (See also December No., p. 333 Par. 99), and Barlow's "Numbers, p. 299.)