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1, 3, 9, 10, 13, 5, 15, 11, 16, 14, 8, 7, 4, 12, 2, 6; by rejecting the multiples of 17. “Whence (by art. 219) the first two periods will be.

PER+R+13+15R 16+R+R*+r?,

p'=R+R10+R+r+R"+R?+R+R Now

p+p=-1; and

pp' =ptptptp+p+p+p+pl=4(p+p=-4. • Hence the quadratic equation containing the roots p,p', will be

pi +p-4=0. - Whence,

p=1+1v 17, and pl=-11/ 17. • Again, the periods of roots pa p', must be now decomposed into the four following periods, the sums of which are, for distinction sake, represented by 9, q'; viz.

Sa RAR1+R 16+r",

= R'+R1+RSTR.

Son! =R* +R5 +R" R Period p2911 =R1+R"+R? +r". . And here we have


99'=' +9+9+9=ptp=-1. • Whence the quadratic equation containing the roots 9,9!, is

q-p=0; consequently,

9=žp+IV (4+p?), and qil=f-(4+pre).

qır!'(4+p:2), and q' =19-V (4 +p.) • Again, the above periods of q, 99, &c., and each decomposed into two periods of two terms each, which new periods are represented by t, t', t'l, &c.; vis. Period into

R? +R 16,
It' = R13+R4

Ro +R®,
Period q', into


Period q", into

FHV =R +",

+12. Period q111, into

trii = R14

. • Now

**=q={p.tiv (4+1%), and

ttl=ti'+x=ql={p+1/(4+p?2.) • Therefore the quadratic equation containing the roots t, t', is .

9+9=0. · Whence,

st'll 19+1 (9-49''),

{t=19-19-49') • The first of these is the greatest positive roots, and is therefore, the value

« In the same way,

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360° of 2 cos. --; which, by substituting for.q and q", their respeetive values, 17

360° in terms of p and p', becomes 2 co$.

{{p+1/(t+p%)} +iv{ip+1 (4+pe)}

4{{p! + 1V.(4+p:2)} • Again, reestablishing the values of p,p', we have in nụmbers, 2 cos. 360°.


i{(+1717)+1V:(17–117)} +


360° which is the true numeral value of 2 cos. -whence it is manifiest, that,

17 by the construction of three quadratic equations, a 17 sided polygon may be inscribed geometrically in a circle.'

After some interesting remarks connected with this curious problem, the author adds,

• Hence we have the following series of polygons, each of which admits of a geometrical construction, viz. Polygons of less than 100 sides, admitting of a Geometrical Construc

tion, No. of Sides. No. of Sides.

No. of Sides. 3 trigon. 16 24

48 3.24 4 2

17 24 +1 51 17.3
22 +1


60 15.22
24 3.23

64 26
8 23
30 15.2

32 25


5.24 12 3.22 34 17.2


17.5 15 5.3


96 • To the above, we may add the three consecutive polygons.

255, 256, 257. each of which is inscribable in a circle ; for

255 = 3.5.17, 256 = 28, and

257 = 28+1, a prime. The next three consecutive polygons, that admit of a geometrical con. struction, are the following ; viz.


255 x 257, 65536 26.

65537 = 2*6+1, a prime.' The extent to which we have carried this critique, will be a proof of the estimation in which we hold Mr. Barlow's performance. We do not mean to affirm that it is free from er



= 17.29

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rors or failures, though we have only detected to in the course of our examination. Of these, the first is in the attempt to demonstrate that a xb=bx a, which fails by reason of too restricted a definition of numbers. The second relates to Fermat's theorem on the impossibility of the indeterminate equation 2 + y = 5", for every value of n greater than 2: the attempt is very ingenious and elaborate, but fails by reducing the proposition to a former corollary which does not apply, and which is not demonstrated, if it did.

To conclude, this work seems well fitted to be read immediately after Euler's Algebra, in some parts of which the investigations are directed to kindred subjects. They who are pleased with the manner of the celebrated German algebraist, will find much in Mr. Barlow's volume to suit their taste. To all indeed who cultivate the Diophantine analysis it will be a great treasure; and eren those mathematicians who rank too high to need its instruction, will be pleased with the judgement and taste evinced by the author in its composition,

Art., III. Memorandum on the Subject of the Earl of Elgin's Pursuits in

Greece. 8vo. pp. 77. Price 6s. Miller. 1811. WHEN Phidias, who made in Athens, and makes in this

tract so distinguished a figure, was performing the pro- . cess under which a rude block of marble was to become a beautiful or majestic human form, he despised no implement or operation, however slight, which could in any manner or degree contribute to the perfection of that intended form. There is in this world, under the denomination of the human mind, a rude and perverse intellectual substance, incomparably harder to be brought to any thing like a perfect shape, than any piece of stone that the artist ever had to work upon. It is, however, under a grand process : and we have sanguire hopes that it will come forth, at length, wrought to a degree of excellence which will contrast, wonderfully and delightfully, with its former condition. This excellence must include, and partly consist in, a highly improved faculty for the general perception of order and beauty, -an intelligence not only of the chief relations and harmonies in metaphysical and moral truth, but also of that kind of rectitude which constitutes order and beauty in the material world. Beyond all question there is such an analogy throughout all the subjects of knowledge, that the faculty of perceiving and admiring the true and the beautiful in higher subjects of contemplation, will be in some certain degree qualified and disposed to perceive and admire them in the inferior classes of subjects. If, therefore, we antici. pate a noble amendment in the general state of the human

mind, we may expect that, along with increasing rectitude : 5 ideas concerning truth in subjects of primary importance, there will be an improvement in the justness of apprehension relatively to the subjects of what we call Taste. And we may justly be gratified that the process is actively and effectually going on in civilized society, for promoting this subordinate part of our intellectual improvements--provided the means be not too expensive, and the measure of time and operation out of all proportion to what is given to much more important matters.

No doubt it would be far the most pleasing to a man with a right comparative estimate of the different parts of that general improvement, toward which it is assumed that the intellect of society is in progression, to see the most forward points of the advance to be in the direction of the improvements that are the most important. He would be extremely happy to see the civilized world making a progress in the wisdom of religion, morals, politics and legislation, with a much slower. growth towards a finished judgement in architecture, sculpture, and painting. Nonwould scarcely any, state of the social mind appear to him more perverted and contemptible, than that in which these refinements of art and taste should be making a distinguished advance, while superstition or scepticism were repressing religion, while a loose moral code extenuated profligacy, and a barbarous legislation was sanctioned or permitted by the prevalence of absurd political opinions.

It were vain, however, to hope, as yet, of such a perverse and frivolous company of beings as mankind, that, even when in a course of improvement, they should give a precedence to the most important pursuits. We must be content to think it, for the present, a great thing, if they are any where making one-fifth part of the progress in religious and political illumination that they do in the cultivation of taste. Let civilized society, or any particular nation, but manifest such a degree of amelioration in respect to the more serious concerns of human nature, as to give unequivocal signs that men are really approaching a considerably higher state of wisdom and virtue, under an impulse that is not likely to remit,-let thus much be realized of the more indispensable kinds of improvement, and it will so far indicate a general soundness of the moral and intellectual system, as to prevent our suspecting the augmented passion for the fine arts to be a kind of exhalation from fermenting moral corruption. Though regretting to see it prevail in a greater degree, and with greater effect, than the zeal in nobler pursuits, we shall yet hope it will not, on the whole, 'counteract that zeal ; and that, though it is operating very prematurely, its effects will ultimately combine with those of that nobler zeal, in the one grand result, the whole improvement of our nature. A philanthropist while thus pleased to see this improvement. (though disproportionate and premature,) of the humsu faculties in one mode of their application-because he anticipates that when at length this too forward atta:ument shall be overtaken by the more important ones, it will tall gracefully into the system of improvements, and be autisfied to hold a very subordinate place in it,--will not, of course, despise the means brought in aid of this subordinate párt of our mental cultivation. Even the foolish extrava, gance of the enthusiasts for the fine arts, who will talk about the more prevailing study, or the improving style, of sculpture and painting, in such magnificent terus as they would have no patience to hear applied to the diffusion of Christianity, or the deliverance of a nation froin an inveterate tyranny, -even this will not provoke him to deny that some small intellectual benefit may be derived,' in England, from delineations of the ruins of Athenian strictures, and from actual fragments of the statues and bas reliefs with which they were once adorned. Put things in their right gradation, from the highest extreme to the lowest, and the man that gratefully exults in our having so long received from Judea, and indeed partly from Greece, the grand rectifier of our intellectual and moral faculties, in their most important relations--the Bible, will not therefore fail to acknowledge the value (though certaivly small according to his scale) of these latest contributions of Greece to discipline our faculties to a more correct perception of beauty in forms.

It is true, that the Christians of the earlier ages, who inhabited the regions enriched with the superb and beautiful works of Pagan art, gave proof, by the zeal with which in some instances they defaced or demolished them, how little they combined with their affection for what instructed them in the most important truth and in their eternal interests, an esteem for what would have so powerfully assisted the formation of a perfect taste, in themselves and their posterity. And, assuredly, it will be doing them no wrong to say, that if they had been possessed, or desirous to be possessed, of so judicious a taste as would be required to constitute a part of that high general cultivation of the mind, which it may be hoped mankind will one day attain, their zeal to destroy these works would have been much more 'restrained. But still, if the Christians, in the time of Theodosius and the following periods, had possessed as fine a taste as the Athenians in the age of Pericles, they must necessarily have beheld the grand and beautiful apparatus of idolatry in a very different light from that in which its remains may now be contemplated. These miracles, as in a

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