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One of the objects of the paper which last year I had the honour of presenting to the Royal Society, was to shew the in­ sufficiency in mathematical reasoning, of a principle of analogy, by which the properties demonstrated for one figure were to be transferred to another, to which the former was supposed to bear a resemblance; and the argument for the insufficiency of the principle was this, that the analogy between the two figures was neither antecedent to calculation, nor independent of it, and consequently could not regulate it; that analogy was the object of investigation, not the guide; the result of demonstra­tion, not its directing principle. Having shewn that analogy could not establish the truth of certain mathematical conclusions, I next endeavoured to shew why such conclusions had been rightly inferred; not by pro­posing any new excogitated principle, nor by pointing out an hitherto unobserved intellectual process; but I conceived they might be obtained by operations conducted in a manner similar to that by which all reasoning with general terms is conducted, and that the relations between the symbols or general terms were to be established by giving the true meaning to the con­necting signs, which indicate not so much the arithmetical computation of quantities, as certain algebraical operations. It was further observed, that, from certain established formulas, abridged symbols or general terms might be formed, which consequently must have their signification dependent on such formulas; and that, although the parts of certain abridged ex­pressions could not separately be arithmetically computed, yet the expressions themselves might be legitimately employed in all algebraic operations.

My Dear Sir, In a Memoir which I had the honour to present to you two years ago, I related some experiments on trees, from which I inferred, that their sap, having been absorbed by the bark of the root, is carried up by the alburnum or white wood, of the root, the trunk, and the branches; that it passes through what are there called the central vessels, into the succulent part of the annual shoot, the leaf-stalk, and the leaf; and that it returns to the bark, through the returning vessels of the leaf-stalk. The principal object of this Paper is, to point out the causes of the descent of the sap through the ’bark, and the consequent formation of wood. These causes appear to be, gravitation, motion communicated by winds or other agents, capillary attraction, and probably something in the conformation of the vessels themselves, which renders them better calculated to carry fluids in one direction than in another. I shall begin with a few observations on the leaf, from which all the descending fluids in the tree appear to be derived. This organ has much engaged the attention of naturalists, particularly of M. Bonnet: but their experiments have chiefly been made on leaves severed from the tree; and, therefore, whatever conclusions have been drawn, stand on very questionable ground. The efforts which plants always make to turn the upper surfaces of their leaves to the light, have with reason induced naturalists to conclude, that each surface has a totally distinct office; and the following experiments tend strongly to support that conclusion.

My Dear Sir, It is well known that the fluid, generally called the Sap in trees, ascends in the spring and summer from their roots, and that in the autumn and winter it is not, in any considerable quantity, found in them; and I have observed in a former Paper, that this fluid rises wholly through the alburnum, or sap-wood. But Du Hamel and subsequent naturalists have proved, that trees contain another kind of sap, which they have called the true, or peculiar juice, or sap of the plant. Whence this fluid originates does not appear to have been agreed by naturalists; but I have offered some facts to prove that it is generated by the leaf; and that it differs from the common aqueous sap owing to changes it has undergone in its circulation through that organ: and I have contended that from this fluid ( which Du Hamel has called the suc propre , and which I will call the true sap,) the whole substance, which is annually added to the tree, is derived. I shall endeavour in the present Paper to prove that this fluid, in an inspissated state, or some concrete matter deposited by it, exists during the winter in the alburnum, and that from this fluid, or substance, dissolved in the ascending aqueous sap, is derived the matter which enters into the composition of the new leaves in the spring, and thus furnishes those organs, which were not wanted during the winter, but which are essential to the further progress of vegetation. Few persons at all conversant with timber are ignorant, that the alburnum, or sap-wood of trees, which are felled in the autumn or winter, is much superior in quality to that of other trees of the same species, which are suffered to stand till the spring, or summer: it is at once more firm and tenacious in its texture, and more durable. This superiority in winterfelled wood has been generally attributed to the absence of the sap at that season; but the appearance and qualities of the wood seem more justly to warrant the conclusion, that some substance has been added to, instead of taken from it, and many circumstances induced me to suspect that this substance is generated, and deposited within it, in the preceding summer and autumn.

When I had ascertained that carbonaceous substances, whether vegetable, animal, or mineral, were capable of being converted into a product, which, by its effects on earthy and metallic solutions, on dissolved gelatine, and on skin, resembled the natural vegetable principle called Tannin , I was at first inclined to give it the name of artificial or factitious tannin ; but some eminent chemists of this country, for whose opinions I have the highest respect, considered this name as objectionable; for although the artificial substance resembles tannin in the parti­culars above stated, yet in one character there appears to be a very considerable difference, namely, the effect of nitric acid; for by this, the artificial substance is, whilst the varieties of natural tannin are destroyed . Such an objection, sanctioned by such authority, induced me to alter the title of my Paper, and to expunge the word tannin wherever it had been applied to the artificial product. In order to satisfy myself more fully on this point, I have, since the communication of my former Paper, made a few ex­periments on the comparative effects produced by nitric acid on those substances which contain the most notable quantities of tannin , and of these I shall now give a succinct account, and shall also cursorily notice other experiments, in which a tanning substance has been produced, under circumstances different, in some measure, from those which have been already described.

Having some years back, when reading the learned Mr. Landen's fifth Memoir, discovered the manner of applying a method of differences, to the species of series, whose sums are there obtained by the help of impossible quantities, and having since extended that application, I now venture to offer it to the consideration of others. The practice of this method, in most cases, appears to me extremely simple; and on that account, I am almost induced to imagine, that they have already been considered by mathe­maticians; indeed since the greatest part of this Paper was written, I met with Euler's Institutiones Calculi integralis ; two simple series are in that work summed by multiplications similar to those employed in the investigation of the principal theorems contained in this Paper; but whether that learned mathematician has farther pursued the method, in that or in any other work, I have not as yet been able to ascertain.

The observations which accompany this Paper were made at Westbury in Somersetshire, in the years 1800 and 1801, with an Astronomical Circle of two feet and a half diameter, constructed by Mr. Troughton, and considered by him as one of the best divided instruments he had ever made; a drawing of it, with a short description, is annexed to the ob­servations. (Plate XX.) When this instrument came into my possession, I thought I could not employ it in a more advantageous manner, than in endeavouring to determine the declinations of some of the principal fixed stars. The various catalogues differed so much from each other, and such doubt existed as to the accuracy of those which were thought most perfect; that the declinations of few stars could be considered as sufficiently well ascertained for the more accurate purposes of astronomy.

The theorems relative to finite differences, given by M. Lagrange in the Berlin Memoirs, for 1772, have much engaged the attention of mathematicians. M. Laplace has been particularly successful in his investigations respecting them; yet an important difficulty remained, to endeavour to surmount which is the principal object of this Paper. The theorems alluded to may be thus stated. Let u represent any function of x . Let x + h , x +2 h , x +3 h , &c. be successive values of x , and 1 u , 2 u , 3 u &c. corresponding successive values of u . Let Δ n u represent the first term of the n th order of differences of the quantities u , 1 u , 2 u &c. And let also S n u represent the first term of a series of quantities, of which the first term of the n th order of differences is u . Then ( e representing the series 1 + 1 + 1/1.2 + 1/1.2.3 +, &c.)

Mr. William Garrard having shewn me a curious property of the tangents of the three angles of a plane triangle, or in other words, of the tangents of three arches trisecting a semicircle, in a paper which I have communicated to this Society, I was led to consider whether a similar property might not belong to the tangents of three arches trisecting the whole circumference; and, on examination, found it be so. Let the circumference of a circle be divided any how into three arches A, B, C; that is, let A + B + C be equal to the whole circumference. I say, the square of the radius multiplied into the sum of the tangents of the three arches A, B, C, is equal to the product of the tangents multiplied together. I shall demonstrate this by symbolical calculation, now commonly called (especially by foreign mathematicians) analytic calculation.

In the first part of this paper, I have pointed out a variety of methods that will give us coloured concentric rings between two glasses of a proper figure applied to each other, and it has been proved that only two surfaces, namely, those that are in contact with each other, are essential to their formation; it will now be necessary to enlarge the field of prismatic phe­nomena, by showing that their appearance in the shape of rings has been owing to our having only used spherical curves to produce them. 35. Cylindrical Curves produce Streaks . As soon as it occurred to me, that the cause of the figure of any certain prismatic appearance must be looked for in the nature of the curvature of one or both of the surfaces, that are essential to its production, I was prepared to expect that if a spherical curve, when applied to a plain surface of glass, produces coloured rings, a cylindrical one applied to the same would give coloured lines or streaks. To put this to the proof of an experiment, I ground one side of a plate of glass into a cylindrical curve, and after having given it a polish, I laid a slip of plain glass upon it, and soon perceived a beautiful set of coloured streaks. The broadest of them was at the line of contact, and on each side they were gradually narrower and less bright. The colours in the streaks were similar to those in the rings, and they were in the same manner changeable by pressure as in them. Their order was likewise the same, if we reckon from the line of contact, as with rings we do from the center; so that these streaks differed in no respect from rings, except in their linear instead of circular arrange­ment.

In this discourse I propose to investigate the attractions of a very extensive class of spheroids, of which the general description is, that they have their radii expressed by rational and integral functions of three rectangular co-ordinates of a point in the surface of a sphere. Such spheroids may be characterized more precisely in the following manner: conceive a sphere of which the radius is unit, and three planes intersecting one another at right angles in the centre; from any point in the surface of the sphere draw three perpendicular co-ordinates to the fixed planes, and through the same point in the surface likewise draw a right line from the centre, and cut off from that line a part equal to any rational and integral function of the three co-ordinates: then will the extremity of the part so cut off be a point in the surface of a spheroid of the kind alluded to; and all the points in the same surface will be determined by making the like construction for every point in the surface of the sphere. The term of a rational and integral function is not to be strictly confined here to such functions only as consist of a finite number of terms; it may include infinite serieses, provided they are converging ones; and it may even be extended to any algebraic expressions that can be expanded into such serieses. This class of spheroids comprehends the sphere, the ellipsoid, both sorts of elliptical spheroids of revolution, and an infinite number of other figures, as well such as can be described by the revolving of curves about their axes, as others which cannot be so generated. In the second chapter of the third book of the Mécanique Céleste , Laplace has treated of the attractions of spheroids of every kind; and in particular he has given a very ingenious method for computing the attractive forces of that class which in their figures approach nearly to spheres. In studying that work, I discovered that the learned author had fallen into an error in the proof of his fundamental theorem; in consequence of which he has represented his method as applicable to all spheroids whatever, provided they do not differ much from spheres; whereas in truth, when the error of calculation is corrected, and the demonstration made rigorous, his analysis is confined exclusively to that particular kind, described above, which it is proposed to make the subject of this discourse. I have already treated of this matter in a separate paper, in which I have pointed out the source of Laplace’s mistake, and likewise have strictly demonstrated his method for the instances that properly fall within its scope. In farther considering the same subject, it occurred to me that the investigation in the second chapter of the third book of the Mécanique Céleste , however skilfully and ingeniously conceived, is nevertheless indirect, and is besides liable to another objection of still greater weight; it does not exhibit the several terms of the series for the attractive force in separate and independent expressions: it only points out in what manner they may be derived successively, one after another; in so much that the terms of the series near the beginning cannot be found without previously computing all the rest. This remark gave occasion to the following paper, in which it is my design to give a solution of the problem which is not chargeable with the imperfections just mentioned: the analysis is direct, and every term of the series for the attractive force is deduced immediately from the radius of the spheroid. As the ellipsoid, which comprehends both sorts of elliptical spheroids of revolution, falls within the class of figures here treated of, I have derived, as a corollary from my investigation, the formulas for the attractions of that figure which are required in the theory of the earth: this paper therefore will contain all that is useful on the subject of the attractions of spheroids, as far as our knowledge at present extends, deduced by one uniform mode of analysis.

Several years have elapsed, since my very highly esteemed friend, now Rev. Doctor Mooney, Fellow of Trinity College, Dublin, presented to the Royal Irish Academy a paper on the Extermination of Radicals from Equations. He has illustrated, by sundry examples, the extermination of quadratic surds. As he has rightly observed, the method is universal. Any number of quadratic surds, independent, or dependent, on each other, may be removed from an equation; because, 1. Any quantity, or factor of a quantity, necessarily subjected to the radical sign, is but of one dimension. 2. This quantity or factor being brought to one side of the equation, while the quantities unaffected with it remain at the other, may, by squaring both sides, be freed from the radical sign. 3. By a repetition of these reductions for each remaining independent surd quantity, any number of surd quantities may be converted into rational.

The term function has long been introduced into analysis with great advantage, for the purpose of designating the result of every operation that can be performed on quantity. This extent of signification has rendered it of essential use, but the various applications of which it admits, and the questions to which it gives rise, do not appear to have met with sufficient attention. I propose in the following paper to present an outline of a new calculus, which naturally results from it. It comprehends questions of the greatest generality and difficulty, and will probably require the invention of new methods for its improvement.

In the year 1772, Lagrange, in a Memoir, published among those of the Berlin Academy, announced those celebrated theorems expressing the connection between simple exponential indices, and those of differentiation and integration. The demonstration of those theorems, although it escaped their illustrious discoverer, has been since accomplished by many analysts, and in a great variety of ways. Laplace set the first example in two Memoirs presented to the Academy of Sciences,* and may be supposed in the course of these researches, to have caught the first hint of the Calcul des Fonctions Generatrices with which they are so intimately connected ; as, after an interval of two years, another demonstration of them, drawn solely from the principles of that calculus appeared, together with the calculus itself, in the memoirs of the Academy. This demonstration, involving, however, the passage from finite to infinite, is therefore (although preferable perhaps in a systematic arrangement, where all is made to flow from one fundamental principle) less elegant ; not on account of any confusion of ideas, or want of evidence ; but, because the ideas of finite and infinite, as such, are extraneous to symbolic language, and, if we would avoid their use, much circumlocution, as well as very unwieldy formulæ must be introduced. Arbogast also, in his work on derivations, has given two most ingenious demonstrations of them, and added greatly to their generality ; and lastly, Dr. Brinkley has made them the subject of a paper in the Transactions of this Society,* to which I shall have occasion again to refer. Considered as insulated truths, unconnected with any other considerable branch of analysis, the method employed by the latter author seems the most simple and elegant which could have been devised. It has however the great inconvenience of not making us acquainted with the bearings and dependencies of these important theorems, which, in this instance, as in many others, are far more valuable than the mere formulæ.

In a former Paper which the Royal Society honoured with a place in the last Volume of their Transactions, I endeavoured to explain the nature of the calculus of functions, and I proposed means of solving a variety of functional equations containing only one variable quantity. My subsequent enquiries have produced several new methods of solving these, and much more complicated functional equations, and have convinced me of the importance of the calculus, particularly as an instrument of discovery in the more difficult branches of analysis; nor is it only in the recesses of this abstract science, that its advantages will be felt: it is peculiarly adapted to the discovery of those laws of action by which one particle of matter attracts or repels another of the same or of a different species; consequently, it may be applied to every branch of natural philosophy, where the object is to discover by calculation from the results of experiment, the laws which regulate the action of the ultimate particles of bodies. To the accomplishment of these desirable purposes, it must be confessed that it is in its present state unequal; but should the labours of future enquirers give to it that perfection, which other methods of investigation have attained, it is not too much to hope, that its maturer age shall unveil the hidden laws which govern the phenomena of magnetic, electric, or even of chemical action. When functional equations containing two or more variables occur, their solution presents still greater difficulties than those we have already considered; the new relations which arise, necessarily require a new notation to distinguish them. I shall endeavour, as far as I am able, to apply or extend that already in use; but, as it is almost impossible in the infancy of a calculus to foresee the extent to which it may be carried, or the new views which it may be necessary to take of it, the notation I have used should only be considered as of a temporary nature; it may be employed until some more convenient one be devised: perhaps, how ever, it might be more advantageous that it should not be altered until our acquaintance with this subject becomes more intimate, and until the infinitely varied and comprehensive relations displayed in the doctrine of functions, have been more minutely examined.

1. I have endeavoured, in this short Paper, to give a simple and connected theory of the construction of logarithms, which I think has not hitherto been done. Prop. I. To find the Logarithm of 1 + x . It is not difficult to see that we may assume L (1 + x ) = ′A x + ″A x 2 + ‴A x 3 + ‴′A x 4 + &c., whence L (1 + y ) = ′A y + ″A y 2 + ‴A y 3 + ‴′A y 4 + &c., and L {(1+ x ) (1+ y )} = L(1+ x + y + xy ), or putting 1+ x = π , =L{1+( x + πy )=′A( x + πy )+″A( x + πy ) 2 +‴A( x + π y) 3 +&c.

In looking into Mr. Spence’s ingenious “Essay on Loga­rithmic Transcendents,” a work published in 1809, but which I have been so unfortunate as never to have seen till within the last fortnight, I was not a little surprised to find that a demonstration of the binomial theorem, similar to the one I had the honour to present to the Royal Society, had been already given by that writer. The same may be said of the first proposition of the preceding Paper on the construction of Logarithms. Having made this acknowledgment, I shall perhaps be pardoned for observing, that Mr. Spence is not particularly happy in the manner of developing the kind of functions he treats of in his preface. I shall endeavour to give the solu­tion of a class of equations of which he (Pref. p. vii.) has considered a particular case: with this we will begin.

The processes which it is the object of this paper to explain, were discovered several years since; but certain difficulties connected with the subject, which I was at that time unable to explain, and which were equally inexplicable to several of my friends, to whom I had communicated these methods, induced me to defer publishing them, until I could offer some satisfactory solution. These observations refer more particularly to the second method which I have detailed in this paper, and which may not inappropriately be called the method of expanding hori­zontally and summing vertically . Some traces of this method may, perhaps, be found in former writers, and particularly in a paper by Professor Vince, “On the Summation of Se­ries,“ printed in the Philosophical Transactions for 1791; but there exists this peculiarity in that which I have employed, that after a certain number of the vertical columns are summed, all the remainder either vanish, or else have some common factor. This method, which I employed about the year 1812, gave the values of a variety of series whose sums had not hitherto been known, most of which were apparently correct, but some of the consequences which followed were evidently erroneous. About this time, Mr. Herschel, to whom I had communicated these anomalous results, by following a very different course, arrived at several general theorems, which, when applied to the series I had obtained, gave the same results. This coincidence at first increased my confidence in the values so discovered, and I continued to examine the reason why my own formulæ were in some cases defective.

Dr. Ure commences this paper by adverting to the fallacies to which the modes of analysing organic substances hitherto practised are subject; and in detailing the peculiar methods adopted in his own researches, he shows the means of obviating them, and of diminishing the various sources of inaccuracy to which these complicated processes of analytical chemistry are necessarily more or less liable. Where oxide of copper is used, its hygrometric quality has generally been overlooked, or not duly allowed for; and the animal and vegetable substances have not in general been exposed to any process of desiccation sufficiently exact or uniform; the author therefore always used the oxide of copper in some known or ascertained degree of humidity; and he dried the organic bodies in the air.pump vacuum, aided by the absorbent powers of a surface of sulphuric acid in the apparatus, and with precautions which he fully describes. He then details the best means of applying heat for the decomposition of organic substances, and describes a drawing representing the construction of his furnace, and other implements. Lastly, he points out the method of examining the results and products, and gives in detail the analysis of sulphuric ether, as illustrating the mode of computing the relations of the constituents, while the results of the other analyses are, for the sake of brevity, thrown into a tabular form. Dr. Ure concludes his paper with some general remarks on the analytical details. In respect to sugar, he observes, that on comparing pure crystalline sugar with diabetic sugar, the latter exhibits a notable excess of oxygen; and he considers weak sugars (as the refiners call them), in general, to exhibit the same peculiarity. In applying the atomic theory to his experimental results, the author enlarges on the different views which may be taken of the ultimate constitution of a variety of organic products, and enters at considerable length into details relating to the vegetable acids, with a view of determining with exactness their prime equivalents, and the relative proportions of combined water which they contain in their crystalline states.

I have recently received from Captain Kater a printed paper from the Philosophical Transactions for 1821, giving an account of his experiments in examining and comparing the different standard scales. I have read with great attention and satisfaction the whole of his results, and am glad to find that the Commissioners for considering the subject of weights and measures, have adopted Mr. Bird's scale of 1760, as by that means there is now a universal standard of comparison, which applies to the French metre, and to all the measures used on the Continent. From Captain Kater’s results it appears, that my standard scale requires a multiplier of —, 000018 to make it agree with the above scale of Mr. Bird; and that Ramsden's bar, used in the Trigonometrical Survey of Great Britain, requires the multiplier +, 00007. That is to say, with respect to a measurement on the meridian, the degree depending on my brass scale must be multiplied by, 000018 and the product subtracted from the measure given by the scale, to reduce it to what it would have been, had it been measured by what is now the parliamentary standard; and the degree depending on Ramsden's bar, must be multiplied by , 00007, and the product added to the measure given by the bar, to reduce it to the standard measure.

It was originally my intention, in entering on this inquiry, merely to ascertain the composition of the ores, the mineralogical characters of which have been so ably delineated by Mr Haidinger in the preceding paper. I had advanced however but a short way in the investigation, when my progress was arrested by doubts both as to the manner of conducting the analyses, and as to the mode of calculating their results. In this uncertainty I found it necessary to extend my original plan, with the view of supplying by my own researches what appeared to be not sufficiently established by the labours of other chemists. I have accordingly divided the essay into two parts; attempting in the first division to ascertain the atomic weight of manganese, and the composition of the artificial oxides of that metal; and in the second, applying the facts thus established to illustrate the chemical constitution of the native oxides described by Mr Haidinger.

The fruit in this natural order does not appear to have been well understood by most botanists; and I shall therefore attempt to give a general view of what appears to me to be its structure; and most of the parts are visible in the section which is here given (Plate IX. Fig. 1.) of the beautiful but insipid Indian Melon (Cucumis Melo) called Phuti.The outer parietes (Fig. 1. a,) when young, are thick, fleshy, and undivided by sutures, with an uniform rind, not separable from the fleshy part. As the fruit ripens, the rind in some cases becomes so thin as to be unable to contain the pulpy matter, and bursts either gradually, as in the melon, or with elasticity as in the Momordica and Elaterium of Tournefort. At other times, the rind hardens either into a thin substance like leather or strong paper, as in the Luffa, or into a strong ligneous covering, as in the Cucurbita leucanthema or gourd. In these cases, it sometimes opens horizontally, by means of an operculum, which falls off and leaves an aperture for the seeds, as in Fig. 2. representing the summit of the Luffa called Picinna in the Hortus Malabaricus.

About three months ago I wrote a paper intitled "Consideration of the objections raised against the geometrical representation of the square roots of negative quantities,” which paper was communicated to the Royal Society by Dr. Young, and read on the 19th of February last. At that time I had only discovered the manner of representing geometrically quantities of the form a + b √— 1, and of geometrically adding and multiplying such quantities, and also of raising them to powers, either whole or fractional, positive or negative; but I was not then able to represent geometrically quantities of the form a + b √ — m + n √ — 1 , that is, quantities raised to powers, whose indices involve the square roots of negative quantities. My attention, however, has since been drawn to these latter quantities in consequence of an observation which I met with in M. Mourey’s work on this subject (the work which I mentioned in my former paper); the observation is as follows: "Les limites dans lesquelles je me suis restreint m’ont forcé à passer sous silence plusieurs espéces de formules, telles sont celles-ci a √ — 1 , a √ — 1 sin (√ — 1) &c., &c., &c. Je les discute amplement dans mon grand ouvrage, et je démontre que toutes expriment des lignes directives situees sur le mêrae plan que 1 et 1.” where a √ — 1 and 1 1 in M. Mourey’s notation signify respectively a ( 1 1 ) √ — 1/4 and ( 1 1 )according to my notation.

In the course of the adjustment and verification of the copies of the Imperial standard yard, destined for the Exchequer, Guildhall, Dublin, and Edinburgh, I discovered a source of error till then, I believe, wholly unsuspected, arising from the thickness of the bar upon the surface of which measures of linear dimension are traced. The difficulties which I experienced, and the remedy which suggested itself upon that occasion, and which was found efficient, are shortly detailed in the account of the construction and adjustment of the new standards of weights and measures of the United Kingdom of Great Britain and Ireland, published in the Philosophical Transactions for 1826. But as the notice there given occupies little more than a single page, and might therefore pass unremarked, I cannot but think that a fact of such importance in inquiries where linear measures are concerned, and which may be sufficient to account for the discrepancies to be found in the experiments of different observers, ought to be placed before the Royal Society in a more pro­minent point of view than that which it at present occupies. I shall, therefore, first extract from the paper alluded to the part to which I refer, and then add an account of such experiments as I have since made on the subject; and describe a scale which I have caused to be constructed so as almost entirely to obviate the source of error of which I am treating.

The following Table is supplementary to those accompanying the paper “On the Mathematical Theory of Suspension Bridges,” printed in the Philosophical Transactions for 1826. It is deduced from the first Table there given, by the plain operations of common arithmetic; but this admits of a far more ready application than the former, to all cases of practical investigation. The first column contains the deflections or versed sines of the curve, expressed in fractional parts of the double ordinate or Span. It is therefore 2 y divided by x , and their reciprocals are added under each.

This peculiarity was first observed in the axillary arteries and in the iliacs of the Lemur tardigradus, which at their entrance into the upper and lower limbs were found to be suddenly divided into a considerable number of equal-sized cylinders, which occasionally anastomosed with each other, and were regularly distributed on the muscles; whilst the arteries proceeding to the other parts of the body divided in the usual arborescent form. Upon prosecuting this inquiry, it was found that the Bradypus tridactylus, and in some measure also the didactylus, has a similar distribution of these arteries.

This paper relates to an improved solution of a problem by which swiftly converging series are obtained, which are useful in computing the mutual perturbations of the planets; and contains some further remarks and improved formulæ for computing the coefficients, by which the arithmetical work is considerably shortened and facilitated.

The light which is the object of the present inquiry must be distinguished not only from that which we derive from the sun, but also from the brightness exhibited by artificial phosphori, electricity, meteors, and other lucid emanations. The principal bodies which afford the light here treated of, are, 1) Some vegetable and earthy substances, such as rotten wood, and peat earth. 2) Marine animals, some in a living state, viz. the Medusa phosphorea , the Pholas , the Pennatula phosphorea , and the Cancer fulgens ; and most of the marine fishes soon after they are deprived of life. 3) Animal flesh in general, some time after the extinction of life. And, 4) Among insects the creeping glow-worm , the flying glow-worm , the fire-fly , and the great lantern-fly . The numerous experiments described in this paper were chiefly made on marine animals, particularly herrings and mackerel, which were exposed either in the air, or in water impregnated with different salts, and of different temperatures, in a dark vault to which the Doctor assigns the name of his laboratory.

The mode of conducting this important survey having been already noticed in the Journals of the Society on various former occasions, it will only be necessary here to state the progress of the operation, which we find has now been carried on over Essex, the western part of Kent, Suffolk, and Hertfordshire, and portions of the counties contiguous to them.

The object of this paper is to show, that we may be assured of the justness and accuracy of conclusions obtained by means of imaginary quantities, without verifying such conclusions by separate investigations, or without inferring their truth from analogy. In the first part the author premises at some length certain arguments, to show that the operations with impossible quantities must have a logic equally strict and certain with the logic that appertains to real quantities, and that the aid obtained by these quantities would be perfectly useless if such conclusions rested only on the frail basis of analogy. The author proceeds next to show that operations with imaginary quantities are by no means mechanical, but that they are conducted according to the rules of strict and rigorous logic; and that, although strictly speaking no proposition concerning them can be true or false, yet, after the demonstrations of certain formulæ for real quantities, demonstrations with impossible quantities may be legitimately and logically conducted. The series, for instance, for the development of an exponential, when the exponent is an impossible quantity, can never, independently of certain arbitrary assumptions, be duly established; and yet, when the exponent is the sign of a real quantity, the formulæ for the development may be rigorously proved. With regard to demonstration, it is shown, as in the case of real quantities, it actually proceeds by a series of transformation, each proved to be the same as the foregoing, not by any arguments grounded on the properties of real quantities, but by reference to the forms certain abridged symbols are made to represent, and to the nature of the operations directed to be performed with certain signs.

In the second volume of the Memoirs of the National Institute, M. Bossut announces a theorem relative to the solidity of a sphere similar to Viviani’s, by which quadrable portions of a hemispherical vault are assigned. M. Bossut withholds the analysis that led him to his result, but mentions that it involves an integration much more complicated than that which occurs in Viviani’s problem. In the present paper, Mr. Woodhouse furnishes the analysis that leads to the result announced in Bossut’s theorem, and, by a transformation of the co-ordinates of the sphere, arrives at a differential expression, the integration of which does not appear more complicated than that employed in the solution of Viviani’s problem.

The author, in the prefatory part of this paper, points out the dif­ference between the two methods of solving problems,—the one using lines and diagrams as the signs of quantity, and making an individual to represent a genus; and the other employing generic terms and signs, which bear no resemblance to the things signified: and insists that, in order to make the process of deduction distinct, exact, and luminous, only one of the two methods ought to be adhered to. This, he says, has not been sufficiently attended to, expressions and formu­las of the two methods having often been blended together, the con­sequence of which has been much ambiguity and paradox; since the true method of combining algebraical formulas cannot be well under­stood, unless we duly attend to their true analytical source and com­bination. To show that the language of algebra need not be infected with the mode of expression adopted by geometricians, and that it is of itself an adequate instrument of argumentation, is the principal object of Mr. Woodhouse’s paper. And he declares that he has entered on this inquiry, not merely for the sake of gratifying specu­lative curiosity, being firmly of opinion that the process of calculation will be much more direct, sure, and expeditious, if it be duly freed from all foreign encumbrances. In order to illustrate and confirm this opinion, he has selected a few cases from those expressions and formulas which are supposed to require for their solution the aid of geometrical theorems, and of the properties of curves.

The importance of the subject handled in this paper, which, as it particularly relates to the process of tanning leather, will be allowed to be of sufficient magnitude, has of late excited the attention of several able philosophers, among whom Mr. Seguin was the first who ascertained the peculiar vegetable matter which is essential to this process, and which is possessed of the characteristic property of precipitating gelatine from its solutions. Mr. Proust has since investigated many other properties of this substance ; but neither these, nor any other chemists, have as yet carried their investigations so far as to determine the various affinities of tannin, and especially how its action upon animal matters is modified by combination with other substances. This task was reserved for our author, who during the two last years bestowed most of his leisure hours on a course of experiments on this subject ; and he here lays before the Society an account of their general results. His chief design was to elucidate the practical part of the process ; but in pursuing it he found himself necessarily led to general chemical inquiries concerning the analysis of the different vegetable substances containing tannin, and their peculiar properties. The paper consists of five parts, the titles of which are as follows :—1. Observations on the analysis of astringent vegetable infusions. 2. Experiments on the infusions of galls. 3. Experiments and observations on the extracts of Catechu, or Terra Japonica. 4. Experiments and observations on the astringent infusions of barks, and other vegetable productions ; and 5. General observations.

After some general observations on the art of embalming, as it was practised by the ancient Egyptians, and on the various kinds of animals embalmed by them, Mr. Pearson proceeds to- give a particular description of the very perfect mummy of an Ibis, which forms the chief subject of the present paper. This mummy was taken out of the catacombs at Thebes, by the late Major Hayes, in the year 1802 or 1803. It was enveloped in cloth, and contained in an earthen jar, similar to those which are found at Saccara. Upon unrolling the bandage with which the mummy was covered, it was found to consist of strips of cloth, about three inches broad, which were strong and firm. The first circumvolutions of this cloth separated easily; but as the work proceeded, they were found to adhere more firmly, and at last were so closely united, that it was necessary to divide them by means of a strong knife. Each layer of cloth seemed to have been imbued with some bituminous substance in a liquid state; and the bandages were further secured by means of thread, in such a manner that the whole mass was rendered firm and coherent; when the bandage was removed, the bird appeared to be covered with the same kind of bituminous substance that had cemented the strips of cloth. As much of this substance as could be removed without injuring the bird was now carefully taken off; and after the labour of several hours, Mr. Pearson succeeded in displaying the whole bird as it had been originally deposited by the embalmer.

Mr. Knight, in the papers formerly communicated by him to the Royal Society, endeavoured to prove that the fluid by which the va­rious parts added to trees, &c. are generated, has previously circu­lated through their leaves, either in the same or in the preceding season, and has subsequently descended through their barks. There is, however, a circumstance stated by Hales and by Du Hamel, which appears to militate against the above hypothesis, namely, that when two circular incisions are made, at a small distance from each other, through the hark, round the stem of a tree, and the bark between these incisions is wholly taken away; that portion of the stem which is below the incisions continues to live, and to increase in size, though much more slowly than the parts above the incisions. The above-mentioned naturalists have also observed, that a small elevated ridge is formed round the lower lip of the wound, which makes some slight advances to meet the hark and wood, projected in larger quantities from the upper lip of the wound. Our author, in a former paper, attempted to explain the above cir­cumstance, by supposing that a small part of the true sap, descending from the leaves, escapes downwards, through the porous substance of the alburnum: in another paper he has shown, from the growth of inverted cuttings, the existence of a power in the alburnum to carry the sap in different directions; and he now describes some ex­periments made in order to show that the conclusions drawn by him are not inconsistent with the facts stated by Hales and Du Hamel; and that although the ascending sap usually rises through the al­burnum and central vessels, yet the alburnous vessels appear to be also capable of an inverted action, when such action becomes neces­sary to preserve the existence of the plant.

This paper is merely an extension of one formerly communicated to the Society by Mr. Robertson, and printed in the Philosophical Transactions for the year 1795. It is, the author says, so far as re­lates to the raising of integral powers, the same as that paper, and is confessedly new only to the extent mentioned in the title, namely, that the present demonstration is applicable when the exponent is a positive or a negative fraction. The nature of the paper is obviously such, as to render it unsusceptible of abridgement.

If, Mr. Manning observes, there existed as full and extensive lo­garithmic tables as ever will be wanted, and of whose accuracy we were absolutely certain, and if the evidence for that accuracy could remain unimpaired through all ages, then any new method of com­puting logarithms would be totally superfluous, so far as concerns the formation of tables, and could only be valuable indirectly, and inasmuch as it might show some curious and new views of mathe­matical truth. But the above kind of evidence is necessarily im­paired by the lapse of time, even while the original record remains, and still more when the record must from time to time be renewed by copies; nor is the uncertainty of copies being accurately taken greater in any case than in that of copied numbers. It is conse­quently useful to contrive new and easy methods for computing new tables, or for examining those we already have; and it is particularly useful to contrive methods by which any part of a table may be ve­rified, independently of the rest; for by examining parts taken at random, we may, in some cases, acquire a moral certainty respecting the accuracy of the whole. Among the various methods of computing logarithms, none, our author says, possesses the advantage of forming them with tolerable ease, independently of each other, by means of a few easy bases. This desideratum, he trusts, the method described by him will supply; being very easy of application, as it requires no division, multiplica­tion, or extraction of roots, and has its relative advantages highly in­creased by increasing the number of decimal places to which the computation is carried.

The same property which at the last meeting was stated by Mr. Garrard to belong to the tangents of any three parts of a semicircle, was in this paper extended to all cases of trisection of the whole circle; but the demonstration of course could not be read to the Society.

Since the degree and manner in which the circulation of the blood depends upon the muscular and elastic powers of the heart and arteries are questions belonging to the most refined departments of hydraulics, the author has already submitted to the Society those general principles upon which he designs, in the present lecture, 1st, to inquire what would be the nature of the circulation if the vessels were as inelastic as glass or bone; 2ndly, in what manner the pulse would be transmitted if the tubes were merely elastic; 3rdly, what actions may be ascribed to their muscular coats; and, lastly, what disturbances are occasioned in different kinds of fevers and inflammations. In order to determine the velocity of the blood in different parts, it is necessary to estimate the pressure by which it is urged forward, and the resistance opposed to its motion. From the experiments of Hales, the pressure may be considered as equivalent to a column of seven feet. In order to calculate the resistance , the author employs the theorems contained in his former communication, and adopts the measurements of Keill and others, for the diameters of the aorta, and of its successive subdivisions. The quantity of blood in the arteries is estimated at nine or ten pounds; its velocity in the aorta about eight inches and a half in a second; while that in the capillary arteries is about 1/93rd of an inch in a second (the diameter of these vessels being about 1/1100th of an inch). The resistance that would be opposed to water circulating under the same circumstances is calculated to be equivalent to a pressure of a column of twenty inches; but the resistance to the motion of the blood is supposed in consequence of its viscidity to be about four times as great, and is consequently stated as eighty inches.

This calculus, which is considered as the largest on record, weighs 44 ounces avoirdupois. It was taken after death from the bladder of Sir James Ogilvie, who had submitted to a fruitless attempt to remove it by the usual operation of lithotomy, rather than prolong an existence extremely miserable from this among other consequences of a blow on his back thirty years before. The stone so completely filled the cavity of the bladder, that it was with difficulty taken out, although there was no real adhesion. Its texture was less compact than that of calculi in general, but agreed in appearance with that species which has been called the Fusible Calculus; and it was ascertained by Dr. Powel to contain the same ingredients as it was composed of, the ammoniacal phosphate of magnesia, with phosphate of lime.

An opinion having been entertained by many persons, that alcohol which has been distilled from wine does not exist ready formed in the liquor, but is generated during the process of distillation, Mr. Brande undertook a repetition of Fabroni’s experiment, on which this opinion is principally founded; but when he added four ounces of dry subcarbonate of potash to eight fluid ounces of port, no alcohol was separated, although some of the same wine had previously been ascertained to yield one fifth part of alcohol by distillation. When the same experiment was repeated on the same wine, to which one seventh part of alcohol had been previously added, still none was separated by subcarbonate of potash: but when so much as one third part had been added, then a very small proportion was found to float upon the surface after it had stood twenty-four hours. When madeira or sherry were employed instead of port, the results were nearly the same. Since the method of Fabroni failed of detecting the presence of alcohol, unless the quantity was very considerable, it became necessary to have recourse to some other method of proving or disproving the presence of it as a product of fermentation; and Mr. Brande conceived, that if it were formed by the heat applied in distillation, the quantity should in that case be different when the same liquor was distilled at different temperatures.

Sir Isaac Newton, who first considered the figure of the earth and planets, confined his view to the supposition of their having been originally in a fluid state; and he conceived them to retain the same figure which they assumed in their primitive condition; and those mathematicians who succeeded him in the same path of inquiry have seldom ventured beyond this limited hypothesis, and have shown, that when a body composed of one uniform fluid revolves about its axis, or even if it consists of several fluids of different densities, its parts will be in equilibrium, and it will preserve its figure when it has the form of an elliptic spheroid of revolution oblate at the poles. But though the supposition of original fluidity of the mass simplifies the investigation, it does not seem to be warranted by what we see of the surface; for in that case, Mr. Ivory observes, the arrangement of all the heterogeneous matters would have been according to their densities; those least dense occupying the surface with gradual increase of density to the centre; whereas, on the contrary, nothing can be more irregular than the density of such solid parts of the earth as come under our observation, and the elevation of continents above the level of the sea, as well as the depths of the different channels which contain the waters of the ocean.

In his second paper, Mr. Ivory investigates the attractions of that particular class of spheroids mentioned in the former; for though it is to these that the theorems of Laplace may strictly be applied, it is liable to the important objection, that the terms of his series near the beginning cannot be found without previously computing all the rest. The analysis of Mr. Ivory, on the contrary, is direct; and every term of his series is deduced directly from the radius of the spheroid. In an appendix to these papers, Mr. Ivory adds some remarks upon a memoir of Lagrange, upon the same subject, published at Paris in December 1809, but which had not till lately been received in this country.

The substance here examined by the author, we are told, was first made known by the celebrated Klaproth. It has been ranked as a distinct principle, soluble in water, but insoluble in alcohol or ether, and convertible, by the action of nitric or oxymuriatic acids, into resinous matter no longer soluble in water, but now rendered soluble in alcohol, by a supposed union with oxygen derived from these acids. Mr. Smithson being in possession of ulmin, sent to him from Pa­lermo by the same person who had furnished M. Klaproth with the subject of his researches, has made various experiments, which lead to a different opinion of its nature from that which has been enter­tained.

In a paper communicated to the Royal Irish Academy by Dr. Mooney, the method of exterminating any number of quadratic surds is pointed out by successively squaring them when brought alone to one side of the equation; and the present is an extension of the same method: first, to all surds whose indices are any integral power of 2, as the fourth, eighth, sixteenth, thirty-second power, &c.; and next to cubic surds, and to any number of surds whose common indices are in any manner compounded of the factors 2 and 3; next to any combinations of surds whose indices do not exceed the number 6, and to as many as three surds, neither of whose indices exceed 12, as well as to various others which cannot be concisely specified.