visible by 9, and Mr. T. says the sum of the digits is 9, or (he should have added) a multiple of 9. Now it is well known that when the sum of the digits of any number is divisible by 9, the number itself is divisible by 9. In the second part of the same proposition, he adds, "And the same will be found to take place when the order of the terms is irregularly changed;" but this was previously demonstrated to be true by your able correspondent O. C. F. No. 287, page 446. last part of the proposition, which is new, as far as I know, may be proved to be universally true, as follows. Let

The

be any number consisting of any number of figures, and y a number consisting of the same number of figures, in an inverted order, or the digits placed in any order whatever also. Suppose a greater than y, and let m be any integral number, then amym will always be divisible by 9. For by O. C. F.'s demonstration, as above quoted, x-y will always be divisible by 9, and am -ym = xm-ly +xm-2 y2 + xm-3 y3+ &c. to ym-1; and since one of the factors is divisible by 9, their product amym will also be divisible by 9.

I am, Sir, &c.

EQUIDISTANT ORDINATE

METHOD.

G.S.

Sir,-One of your correspondents expressed a wish, a few weeks ago, to see, in your excellent Magazine, an investigation of this method, upon simple and obvious principles. I have waited some time, under the hope that some of the more scientific contributors to your pages would execute the desired task; but as nothing to this end seems yet to have been offered to your notice, I beg leave to present the following.

Your correspondent seems to wish for something less recondite and elaborate than the elegant method exhibited by Dr. Hutton in his "Mensuration; something which shall be convincing to those, probably, whose theoretic knowledge does not extend beyond the principles of conic sections; and this is now attempted.

• Since the receipt of this paper, we have received another excellent one on the subject from S. S., to which we shall also give a place in an early Number.-ED.

Let ab cd ef be three equidistant diameters of a parabola, and let them be cut at right angles by the straight line ace then the area of the space abdfea, comprehended between the curve bdf and the straight lines ba ae ef, is (ab+4cd + ef) × } ac.

For, drawing through d the tangent st parallel to the double ordinate bf, it is well known (see Hutton's Course, Parabola, th. 16,) that the parabolic space bfd is of the parallelogram bfts. Now, area of the trapezoid gets =(as+et) ac2dc. ac (Hutton, Parab. th. 11). And, area of the tra pezoid acfb (ab + ef) ac. Therefore, area aefb + 2 area aets = (4dc + ab + ef) ac. But, area aefb + 2 area acts 3 area aefb + 2 area bfts 3 area aefb + 3 parabolic area bdf (Hutton, th. 16) 3 mixtilineal area abdfea.

Ŏr, area abdfea -Q. E. Ď.

} ac.

(ab + 4dc + ef)

The demonstration, conducted upon the same principles, would have led to the same result, if the curve bdf had presented its convexity to ac. We may proceed to the

General Proposition.

Let bdfhkmp be a continuous and not very irregular curve, presenting either its concavity or its convexity to a right line an: let ab pn be ordinates drawn perpendicularly from an to meet the curve; it is proposed to measure the plane surface anpm, &c. db by the method of equidistant ordi

nates.

h k

n

Let the base an be divided into an even number (as 6, 8, 10, &c.) of equal parts, and let parallel ordinates cd, ef, &c. be drawn at the several points of division. Then, by the lemma, supposing the portions of the curve to be nearly parabolic ; —

Area aefdb Area eikhf Area inpmk

=

=

=

(ab + 4dc + ef) ao: (ef + 4gh + ik) ac: (ik + 4lm + np) } ac: Consequently, by adding together these three equations, area anpm, &c. db =(ab+4dc+2ef + 4gh+2ik + 4lm +np) ac.- Q. E. F.

Hence, if f and denote the first and last ordinates; e the sum of the even ordinates, as of the 2d, 4th, 6th, 8th, &c.; o the sum of the odd ordinates, as of the 3d, 5th, 7th, &c., not reckoning in them f and 1; d distance between two succeeding ordinates; we shall have-Area anpm &c. db = } d (f + 1 + 4e + 20). This agrees with Rule 4, p. 86, Mensuration of Surfaces, in "Emerson's Course of Mathematics," given there without demonstration; also with the rule at p. 201, Gregory's "Mathematics for Practical Men," given there also without demonstration.

Yet this, after all, it will be remembered, is but an approximation; though, when employed under proper restrictions, a very correct and useful one. Its application to the mensuration of solids may, perhaps, be resumed at a convenient opportunity.

I am, Mr. Editor,

Yours, &c. P. M. W.

[P. M. W., after he had forwarded to us the preceding letter, happening to turn to Part IV. of Hutton's Mensuration, where that admirable author treats so fully on the quadrature and cubature of curves in general, found, what he had in truth forgotten, (not having looked to that part of Hutton's Treatise for some years,) that the first proposition in section 2 of the said Part IV. is essentially the same as the general proposition in this communication, and made to depend upon a similar theorem. The said theorem, or lemma, is, however, demonstrated much more simply than in Hutton's work, so that the communication may retain its utility; although, as P.M.W.

wishes us explicitly to state, it can now have no pretensions to originality.]

THE SPIRIT OF KNOWLEDGE. (Extract from an address by Thomas Steele, Esq. to the Juvenile Liberal Club of Clare.)

A celebrated treatise has been writ

ten upon "The Spirit of Law;" I do not know that any one has yet written a treatise upon "The Spirit of Education," but it would afford a splendid subject expanded into its details. The germ of the subject has been so nobly treated by the Roman historian, Sallust, that I do not recollect any thing which I have ever read to exceed it in justness of sentiment, or to equal it in the concise energy by which the sentiment is struck home to the heart of the reader. It is a succession of "æternæ veritates," put forward with the utmost rapidity and clearness; and the subsequent histories form an exquisite commentary upon such an exordium. What says the Roman historian? He says, "It is right and fitting that all men, who aspire to excel the other animals, should not only exert themselves, but that they should strain their energies (summa ope niti decel) to effect this purpose; and that they should not pass through existence as if they were beasts of the field, which nature has formed with heads not erect, and rendered subservient to the impulses of mere animal nature." He speaks of the brief duration of our personal existence, and urges the consequent necessity of making every effort to cause that our memory may live with posterity as long as it is possible. Now, how does this great Roman historian describe virtue, and how does he describe the nature of the pursuits that ought to occupy the attention of an intellectual being? Of virtue, he says that it is refulgent and eternal (virtus clara æternaque habetur); and, after saying that he estimates alike the life and death of the debauched and slothful, and the ignorant and uncultivated the following is his description of a living man :-" But that man appears to me indeed to live, and to be an 'intelligential substance,'

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who, with his mind exalted in the pursuit of some important purpose, and disdaining all trifling pursuits and dissipation, (Negotium, quasi nec otium,) aspires to fame either by the achievement of some glorious action, or by the exercise of some art useful to society." [Verùm enimverò is demùm mihi vivere atque frui animá videtur, qui, aliquo negotio intentus, præclari facinoris aut artis bona fumam quarit.]

NOTES OF A COURSE OF LECTURES

ON MECHANICS, DELIVERED BY DR. LARDNER AT THE LONDON UNIVERSITY-SESSION 1828-29. Previous to entering into the science of mechanics, it will be necessary to explain a few of the properties of bodies.

Cohesion is that property by which the several particles of any substance, &c. are kept united, and without which there would be no difference in the natures of bodies. All the various phenomena of a body, being tough, hard, soft, &c. are the effects of a greater or less degree of cohesion. Some bodies have greater intensity of cohesion, in proportion to the magnitude of their spheres of action, than others. For example, if a piece of iron be drawn asunder (which will require an immense force) it will be almost impossible with any weight to unite the pieces, however large may be their spheres of action, shewing that the cohesion is but small; while a piece of India rubber may be stretched out to a great length, and, if broken, can easily be joined by pressure, though the sphere of action be but small; by which it may be easily seen how much stronger the cohesion is in India rubber than in iron.

Cohesion exists not only between materials of the same kind, but also between those of different kinds, as between solids and liquids, &c.

There is another kind of cohesion called capillary attraction, which is a propensity that water or any other liquid has to ascend in small tubes, as the pores of sponge, &c. The cause which produces this effect has occasioned great dispute; but the investi

gation of the merits of each opinion is too much mixed up with mathematics for the present course of lectures. An illustrative and amusing experiment may be tried by immersing one end of a piece of thick cotton in a glass of water, and letting the other end hang down on the outside, when the glass will be emptied in a short time.

· Impenetrability is an universal property of matter, and susceptible of no degrees.

Example. Suppose two cannonballs were proceeding towards each other with the same force and velocity in opposite directions, if they were penetrable they would pass through each other and continue their courses; but this is impossible, and therefore they are impenetrable.

The pores of a body are the interstices between the particles which constitute the mass or quantity of matter; and it is by the closeness of these particles that the different degrees of density are formed. Thus, the denser a body is the fewer pores it has. If a tube be half filled with water, and then gently filled up completely with sulphuric acid, after shaking them together, the mixture will be found to have diminished considerably, which may thus be accounted for: The particles of the sulphuric acid being less than those of the water, when mixed together the former Occupy a great part of the space previously held by the pores of the water, in the same manner as a mixture of shots and bullets will occupy less space than the two separately. Silver and copper, when mixed, will produce the same effects, and many persons have been deceived, in mixing spirits and water, by the same phenomena resulting.

The volume of a body means its pores included with its particles.

There are many curious examples of the property of divisibility, which may not be uninteresting. If a pound of silver be melted together with a grain of gold, every grain of the mixture will contain a visible particle of the gold, which may be detached by means of aqua fortis; consequently, one grain of gold may be divided into 5761 visible parts. A grain of musk

may be placed in a room where the air is continually being changed, and at the end of 20 years no sensible - diminution of bulk will be discovered, although it may be supposed that particles of it were constantly flying off to scent the air. Lewenhoek has discovered many other properties which may not be uninteresting. The size of the animalculæ in water bears about the same proportion to a mite that a bee would do to a horse. The milt of one cod-fish contains a greater number than there are human beings on the earth.

It is in consequence of this property that it is impossible there can be a perfect vacuum in the receiver of an air-pump; for though there are some machines which will extract ths of the air contained in them; yet they will be as perfectly filled by the remaining th as they were by the whole.

There are two ways of shewing to how great an extent one may imagine this property to reach.

From a point c in a straight line,

ab, describe the circle agh; to this circle draw the tangent de, and from the centre c draw the line cf, which is the line wished to be divided, and which is cut in the point m. Now, from a point n, still lower in the line ab, describe another circle ail touching the tangent in the same point a as the other one did, and the line cf is again cut by this circle in the point o. Therefore, the part fo is evidently less than fm. Now it is evident, that by increasing the radius of the circle, it shortens the part of the

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To the line ab draw the two perpendiculars cd and ef, and along the line bf mark off even spaces, as at 1, 2, 3, 4, &c., and from the point c draw the lines c1, c2, c3, c4, &c. Now it is evident that the line c2 cuts the line ab at a point nearer a than the line c, and c3 nearer than c2, and c4 than c3, &c.; and as cd and ef are parallel, the lines c1, c2, c3, c4, &c. may be continued to infinity, and each succeeding line will approach nearer to the point a than the preceding one, but will never touch it.

The expression "infinity of division" must not be confounded with "infinity of divisibility;" the former means that degree of infinity which may be attained by real measurement, while the latter implies that to which there are no limits attached.

Previous to concluding this subject, it may be entertaining to describe Dr. Wollaston's ingenious manner of procuring fine wire for astronomical purposes for ascertaining the distance between any stars, &c.; on the accuracy of which depends all the perfection of the study of astronomy. Ithad been found that when the very finest thread was placed across the end of a telescope, it was magnified beyond the star's apparent size, and consequently hid it from view; for although the stars appear very large when seen with the naked eye, yet this is the effect of radiation alone; and a star, when viewed through a good telescope, is nothing but a brilliant point. Dr. W.'s manner of obtaining the wire was this:He procured a slender cylinder of silver, through the axis of which he drilled a hole, the diameter of which was th part of that of the silver;