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cosine of the declination, and the sum will be the sine of arc first.

2. Add together the secant of arc first, and the sine of the declination, and the sum will be the cosine of arc second; which will be acute when the latitude and declination are of the same denomination, but obtuse when they are of different ones.

3. Add together the cosect. of arc first, the cosine of half the sum of the altitudes, and the sine of half their difference; and the sum will be the sine of arc third.

4. Add together the secant of arc first, the sine of half the sum of the altitudes, the cosine of half their difference, and the secant of arc third, and the sum will be the cosine of arc fourth.

5. When the zenith and elevated pole are on the same side of the great circle passing through the places of the sun at the times of observation, the difference of arcs second and fourth will be arc fifth, otherwise their sum will be arc fifth.

6. Add together the cosines of arcs third and fifth, and their sum will be the sine of the latitude.

Note. When the declination and latitude are nearly equal, and of the same name, it may sometimes be doubtful whether the sum or difference of arcs second and fourth ought to be taken for arc fifth. But the computation is soon made on both suppositions; for cosine of arc fifth is the last logarithm which is taken from the tables, and the other parts of the calculation are therefore not affected by the change. One of the results must certainly be the required latitude, and the latitude by account will generally be sufficient to determine which of them ought to be taken. But when the sum of arcs second and fourth is equal to 90°, or greater, it can only be their difference which is arc fifth.

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Example.-Given the altitudes of the sun 19° 41′ and 17° 13′, interval one hour, and the sun's declination 20° S. to find the latitude, it being by account about 50° N.

19° 41'
17 13

Sum 36 54,

Diff. 2 28,

half sum 18° 27′ half interval 7° 30′
half diff. 1 14

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I have introduced the rule in the above form among the young navigators in this Institution; and if what I have said may be the means of drawing the attention of practical men to the subject, my object in writing these observations will be attained.

XXVIII. On the Relation of Acids and Alkalis to vegetable Colours, and their Mutations thereby. By JOHN MURRAY, F.L.S. M.G.S. M.W.S. &c. &c.

To the Editors of the Philosophical Magazine and Journal. July 9, 1822.

GENTLEMEN,-You had the goodness to insert in a former Number of the Philosophical Magazine (volume lviii. p. 273) a few remarks of mine on the change of vegetable colours by metallic salts; and it is more than a twelvemonth since I have in my prælections pointed out that the mere change of colour produced by acids and alkalis afforded no certain index of their nature. I showed the action of salts of iron, &c. on syrup of violets, &c. in my experiments at the Surry Institution, both in my public discussions and to several individuals in the laboratory of that Institution.

At page 274 of the Philosophical Magazine, vol. lviii. my language, one would think, is emphatic enough. "It seems

evident

evident that we have yet to learn the invariable characteristics of alkalis and acids. We may attempt to cover our ignorance by a free use of the term anomaly; but I do hold that in the universe of God there is no such thing as anomaly."

After all this, I do own that Mr. Faraday's paper on the same subject in Mr. Brande's Journal for 1st instant, pp. 315 and 316, surprised me not a little, because I must assume egotism, though it may appear, that the very humble priority of requesting the attention of chemists to this important fact is due to me, as clearly demonstrated by the paper published in your 58th volume.

Mr. Faraday, however, is not pleased either to advert to that paper or my name as connected with the subject, and his paper concludes in the following language:-"My object has not been to trace these changes as far as possible, but merely to show their general appearance, and to guard against any deceptive conclusion, with respect to solutions tested by Turmeire, and to call attention to the distinguishing characters of acids and alkalis." In the present very imperfect state of chemical science we may not be able" to trace these changes as far as possible;" and I had already called attention to the deceptive conclusions in question. Mr. Faraday has in a former part of this paper observed, "I find on trial, however, so many substances possessing this property," namely, what I had pointed out in the action of subacetate of lead, nitrate and sulphate of copper, &c. " that," continues this ingenious chemist, "it must either be limited more exactly than has yet been done, or else given up as a distinguishing property." How "limited more exactly?" It must be given up entirely.

Mr. Faraday endeavours to account for the change, but the success of this attempt does not seem quite so evident; for in one part of his memoir he thinks that it is in consequence of the protosalts becoming persalts, though he states "submuriate. of zinc," &c. " appeared alkaline to turmeric paper." Further on, it is stated that "the effect is produced principally by the acid present." How can this be, in the case of subacetate of lead? Does this too become a persalt? But he afterwards remarks that "the whole substance must act."

Now the truth is, that all that was done before my experiments, consisted in Desfosses showing that boracic &c. acids reddened turmeric paper; and Mr. F. thinks that South had found that subacetate of lead reddened turmeric paper. It may be so, for any thing I know to the contrary; but I have never met with the detail. From all this however it was merely supposable that turmeric, as turmeric, was susceptible of this change by the action of boracic acid, &c. But I have ad

Y 2

verted

verted not only to turmeric, but to other vegetable colours, as Syrup of Violets, Tincture of Cabbage, Columbine, &c.

In the "Transactions of the Royal Society of London," Mr. Smithson has ingeniously supposed that the transit from red to blue, exhibited on the rupture of the leaves of the red cabbage, &c. is owing to the escape of carbonic acid gas. This however is most certainly not the case. I am induced to ascribe it to the loss of a portion of its latent caloric, for the following reasons: If the vessels be ruptured in a heated atmosphere, the colour continues red. If the blue infusion be heated, it becomes red, which is completely fatal to Mr. Smithson's conclusion. The Tincture of Cabbage will even change to red in a tube hermetically sealed.

Simply heat blue litmus paper or unsized paper tinged with Tincture of Cabbage, and such will become red; and the original blue colour will be restored on dipping the paper into diluted alkali.

Let paper stained with Tincture of Cabbage, and subsequently reddened with acetous acid, be exposed to the effects of radiation in an unclouded nocturnal sky, and the blue colour will

return.

I found that a portion of ice well washed with distilled water, turned the Tincture of Red Beet to a permanent dark brown.

In fact, I have found that heat, in numerous cases, has superinduced a change of colour similar to that effected by acids, and cold similar to those changes connected with alkaline action.

These facts clearly prove that we have been completely wrong in the lines of demarcation we have drawn around alkalis and acids in their relations to vegetable colours as characteristic of their respective features; and I cannot doubt that a more subtile and refined chemistry, the result of an extended stage of our beautiful science, will reveal to us many more blots;blots that the hand of experiment only can deface.

With every respect I have the honour to be,

Gentlemen,

Your much obliged and very obedient humble servant,
J. MURRAY.

Aug. 2, 1822.

I believe (for in truth I very seldom read over what I have written, certainly never take or retain a copy of my communications) I omitted to note, "on the relation of acids and alkalis to vegetable colours," that the superacetate of lead turns the Syrup of Violets green, the same change being superinduced, as in the case of subacetate of lead;-there is therefore no necessity for ringing changes on that solitary string.

It may be interesting to mention, in reference to the siliceous calculus I have described to you, that the urine of the individual (who voided the fragments of which silica formed a constituent part) after a repose of nine days had deposited (spontaneously) a white sediment with mucus superimposed, and much urea beautifully crystalline.

At page 332 of Mr. Children's very excellent translation of Berzelius on the blowpipe (and which adds to the high claims Mr. Children, as an accomplished chemist, has on our respect and gratitude) "siliceous calculi" are described as leaving per se "an infusible, sometimes scoriaceous ash, which fuses with a small quantity of soda, slowly and with effervescence, into a more or less transparent glass globule.”

It is respectfully submitted, as a query, whether the following, page 330, be sufficiently explicit to prevent the obtrusion of error?"The slight ammoniacal odour which potassa develops with almost all animal substances has nothing to do with this," viz. that evolved in the case of calculi composed of urate of ammonia. I always am, gentlemen,

Your obliged and obedient servant,

J. MURRAY.

XXIX. New Demonstrations of the Method invented by BUDAN, and improved by others, of extracting the Roots of Equations. By Mr. PETER NICHOLSON*.

IF A-1
+Bxn-2

and Ax" +Bx"−1

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+Cx+&c=P2"-1 + Qu+ Ro

+Cx"

n-2

+ &c.

'+&c=P'v" +Q'v”-1+R ́v=2. +&c. and if x- ·e=vor x=ve, then will the coefficient of the nth term of the second side of the second equation be equal to the product of the coefficient of the (n-1)th term of the second side of the first equation, and the given quantity e plus the coefficient of the nth term next following.

That is, P' P, Q'= Pe+Q, R'=Qe+R, S'= Re+ S &c.

For by hypothesis,

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n-2

n-3

Demonstration.

n-1

n-2

+Bx+Cx+&c. Po"
'+&c. = Po +Qo'
+ Rrn-s

+&c.

Multiply the first side of this equation by x, and the second

side by its equal o+e, and we have the equation

* Communicated by the Author.

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