the number of assistants necessary were impediments in the way of its being utilised for regular observation, and he assures us he "made it a rule never to employ a larger telescope when a smaller will answer the purpose." It is certain that the mirror which was in the tube in October, 1789, the month following that in which Herschel dates the completion of the telescope, was of excellent definition. On the 16th of that month he followed the sixth and seventh satellites (Enceladus and Mimas) up to the limb of the planet, and witnessed their occultation. Holden writes: "I have never seen so good definition, telescopic and atmospheric, as he must have had on these occasions."

Between the years 1796 and 1799 Herschel made an elaborate classification of stars visible to the naked eye according to their comparative brightness, which he communicated to the Royal Society in four papers published in the Phil. Trans. It formed the first general catalogue of the kind, exhibiting the exact state of the sky in his time. A reduction of Herschel's observations was undertaken by Mr. C. S. Peirce, and the results appear in vol. ix. of the Annals of the Observatory of Harvard College. So far as we know, their reduction had not been previously attempted. Instances of variability in the light of nakedeye stars were detected during the progress of the classification, the most notable discovery in this direction being perhaps that of the periodical fluctuations of a Herculis, in about sixty days. Another star in the same constellation he considered had totally disappeared in 1791, though he had seen it distinctly in 1781 and 1782.

Herschel was led to his numerous discoveries of double stars by his expectation of being able to determine the parallaxes of stars from measures made at opposite seasons of the year of the distances of pairs which appeared near together, and in the search for such pairs, his first catalogue of upwards of 200 double stars was formed and presented to the Royal Society in 1782. Long had previously measured stars upon a similar plan without success, but Herschel pointed out that his stars were not well chosen.

For the successful application of the method it is necessary that one of the pair of stars should really be situated at a much greater distance from us than the other, and as the most reasonable test of distance, Herschel assumed their difference of brightness, so that he sought for pairs where the components differed widely in this respect. The view therefore which he adopted at this time with respect to two stars seen in close proximity to each other was that one was in nearly the same line of sight as the other, but might be far more distant, thus constituting together what we now term an optical double star. From this beginning he was led to the discovery of revolving double stars, stars changing their relative position from year to year; and in 1803 he communicated to the Royal Society his memorable paper: "An account of the changes which have happened during the last twenty-five years in the relative situation of double stars, with an investigation of the cause to which they are owing." He was then satisfied that there were in the heavens pairs of stars which were physically connected with each other. The research for stellar parallax was not successful, but in place of it he discovered the existence of binary systems. He could not in his day decide

whether the motions of suns round suns was obedient to the laws of gravitation, but five years after his death the French astronomer Savary proved that one of these revolving double stars, discovered by Herschel, & in Ursa Major, really was subservient to that law, and as every student of astronomy will be aware, the number of physically connected systems where the elements of the orbits have been determined, is now a large one, and is gradually increasing.

Following at present the order in which Prof. Holden refers to the scientific labours of Herschel, we now arrive at his researches on planets and satellites, respecting which the improvements he made in the construction of telescopes enabled him to advance knowledge so greatly. He was not particularly occupied with the inferior planets, but he determined the time of axial rotation of Mars with

greater precision than before, and also the position of his axis. The times of the rotation of the satellites of Jupiter were found from observations on their changeable brightness, and Herschel also remarked the as yet imperfectly explained phenomena attending the transits of the satellites across the disk of the planet. Saturn, as Holden remarks, was the object of his constant attention: in addition to the discovery of the interior satellites Enceladus and Mimas, he left upon record an extensive series of observations of the seven attendants upon Saturn at that time known, and determined the time of rotation of the outer satellite Japetus upon its axis, by similar observations to those made upon the satellites of Jupiter. He ascertained the time of axial rotation of Saturn, and was the first who had succeeded in effecting this in a reliable manner. He also remarked the curious square-shouldered appearance which the globe of the planet has been suspected to present, and of which we still occasionally hear, though it was long ago proved by Bessel to be an illusion. It is remarkable that notwithstanding Herschel's frequent scrutiny of the planet, with all his experience of observation and the advantages of optical means surpassing by far those of his contemporaries, he does not appear to have at any time suspected the existence of the interior obscure ring. He proved beyond doubt that Uranus was attended by two satellites, and believed he had observed four others, and for a long time on his authority the planet was credited with six attendants.

In 1795 Herschel communicated to the Royal Society a memoir upon the nature and construction of the sun and fixed stars. As to the former he adopted a modified view of the theory which had been advanced by his friend Wilson of Glasgow; he regarded the sun as consisting of three essentially different parts: a solid and non-luminous nucleus, cool and perhaps capable of habitation, above it the atmosphere proper, and still higher the clouds or bodies which cause the sun's intense brilliancy. In this paper occurs a remark which, as Prof. Hoiden observes, has often been brought to bear, in consideration of the causes which maintain the solar light and heat. "Perhaps," he says, "the many telescopic comets may restore to the sun what is lost by the emission of light." We know that however credible in his day points in his theory have given way under our greatl advanced knowledge.

One of the discoveries, or perhaps we should rather say

demonstrations, which especially mark his powers of research and reasoning, was that of the motion of the sun and solar system in space and the direction of this translation, which, considered generally, has received confirmation from more recent and refined investigation. Maskelyne had determined the proper motions of a limited number of the brighter stars, and Lambert, Mayer, and Bradley had thrown out ideas upon the subject, and, following up their suggestions, he showed that the sun was really in motion towards a point in the constellation Hercules, and assigned "the apex of solar motion" with what Holden considers an astonishing degree of accuracy. His second paper on this subject (1805) his biographer views as "the best example that can possibly be given of his marvellous skill in reaching the heart of a matter, and it may be the one in which his philosophical powers appear in their highest exercise."

To gain a knowledge of the "Construction of the Heavens," as Herschel termed it, of the laws of distribution of the stars generally, the star-clusters and nebulæ in space, was confessedly a main object of his astronomical labours, and the memoirs bearing upon this subject extend over the whole period of his scientific career. For this purpose he adopted a system of star-gauging, which in practice consisted in pointing his 20-feet reflector towards various parts of the sky and counting the number of stars in a field of view 15' in diameter. In this way, by methodical observation, the great differences in number of the stars in certain portions of the sky over those in other directions were reliably defined, and in extreme cases the difference was very marked, as in one mentioned by Holden, where in R.A. 19h. 41m., N.P.D. 74° 33', in the constellation Sagitta, the number of stars per field was found to be 588, while in R.A. 16h. 10m., N.P.D. 113° 4′ in Scorpio it was only 1'1-" ein Loch im Himmel!" In this part of his review the author briefly touches upon the views entertained by Herschel at various periods between 1784 and 1817; he considers that while at the commencement of his researches the whole subject was in utter confusion, as they progressed data for the solution of some of the most important questions were accumulated, and the results of Herschel's whole labours form the groundwork upon which future investigators must build. He is the founder of a new branch of astronomy."

The researches for a scale of celestial measures, on light and heat, &c., on the dimensions of the stars, on the variable emission of light and heat from the sun, are briefly referred to. Herschel's observations on the spectra of the fixed stars have been, we believe, very much overlooked. In his memoir in the Philosophical Transactions for 1814 he mentions that in 1798 he made some experiments on the light of a few of the stars of the first magnitude, by a prism applied to the eye-glasses of his reflectors, adjustable to any angle and direction, with the following results:-The light of Sirius consists of red, orange, yellow, green, blue, purple, and violet; a Orionis contains the same colours, but the red is more intense and the orange and yellow are less copious in proportion than they are in Sirius. Procyon contains all the colours, but proportionally more blue and purple than Sirius. Arcturus contains more red and orange, and less yellow in proportion than Sirius. Aldebaran contains much

orange and very little yellow. a Lyra contains much yellow, green, blue, and purple." Holden suggests that if we were to attempt to classify these stars by Herschel's observations alone we should put Sirius and Procyon into one type of stars, which have all the colours in their spectra; Arcturus and Aldebaran would represent another group, with a deficiency of yellow and an excess of orange and red in the spectrum; a Orionis would form a type of those stars, with an excess of red and a deficiency of orange; and a Lyra would represent a sub-group of the first class. The correspondence with Secchi's types and representatives is almost complete.

There remains one other great section of Herschel's researches and discoveries, that relating to the nebulæ and clusters of stars. When he commenced his observations in 1774 very few of these objects were known. Messier's catalogue of sixty-eight such objects did not appear till 1784, and they were chiefly objects found in his long-continued search for comets. Lacaille contributed twenty-eight from his observations at the Cape of Good Hope. Herschel discovered more than 2500, which he distributed in classes as follows:-Class I. "Bright nebulæ" (288 in all); II. "Faint nebulæ❞ (909); III. "Very faint nebula" (984); IV. "Planetary nebulæ " (79); V. "Very large nebulæ" (52); VI. "Very compressed and rich clusters of stars" (42); VII. "Pretty much compressed clusters" (67); VIII. "Coarsely scattered clusters" (88). In addition he pointed out large spaces of the sky covered with very diffused and faint nebulosity, which do not appear to have been re-observed. Holden advises that they should be sought for with a powerful refractor, which would be less open to illusions than Herschel's reflectors, and that the instrument should be used in the way he adopted-in sweeping.

Throughout Prof. Holden's interesting memoir there is evinced the same enthusiastic admiration of Herschel and his scientific labours, and he concludes in the same strain. "He was born with the faculties which fitted him for the gigantic labours which he undertook, and he had the firm basis of energy and principle which kept him steadily to his work. As a practical astronomer he remains without an equal. In profound philosophy he has few superiors."

Lists of Herschel's scientific memoirs and of works bearing upon them, are appended to the volume which has formed the subject of our notice, and which, if it has a fault, is of only too limited extent to do full justice to a long life of discovery and research. We will reiterate the hope expressed by Prof. Holden in his preface, as we understand it, that some member of Sir William Herschel's family may at no distant period "let the world know more of the greatest of practical astronomers". . . "of a great and ardent mind whose achievements are and will remain the glory of England;" and in this connection, that whatever may be found amongst his manuscripts (and as regards the drawings of the nebula, no less an authority than the late Prof. D'Arrest has expressed a strong hope of further publication) may at the same time be given to the astronomical public.1

J. R. HIND

1 Prof. Holden's work is published in London by Messrs. W. H. Allen and Co.

BRITISH FISHES

Natural History of British Fishes: their Structure, Economic Uses, and Capture by Net and Rod. Cultivation of Fish Ponds, Fish suited for Acclimatisation, Artificial Breeding of Salmon. By Frank Buckland, Inspector of Fisheries. (London: Society for Promoting Christian Knowledge.)

IT

T would have been difficult for Mr. Buckland to produce a dull book on any question connected with the economy of our fisheries; his merit in this respect has tended, however, to lead him too much in an opposite direction. It is painful, now that we are deprived of the living presence of the genial naturalist and industrious fishery inspector, to write an unkind word regarding any branch of his life's work; but of this book we are compelled to say that we would have appreciated it better had it been less "familiar" and more scientific. That it should be full of interesting information about fishery matters was quite to be expected from the richness of the stores which its author always had at his command, and if Mr. Buckland had taken pains to digest the matter so lavishly extracted from Land and Water, and had likewise collated the miscellaneous information contained in the volume with care, he might then have enjoyed the satisfaction of presenting to the public a natural history of British fishes which probably would have compared satisfactorily with other good books of the kind. It is not too much to affirm that a carefully edited selection from the numerous essays contributed to the various blue-books to which the deceased gentleman was so voluminous a contributor, would have made a more interesting volume than the present work. The fact is, Mr. Buckland was nothing if he was not sketchy and rapid; he would not be tied down to severe statements, but preferred to give an off-hand opinion in a dashing way, no matter that he might find out within the year that what he had advanced was very far wrong. In the present volume, as a glance at the plethoric title-page will show, Mr. Buckland attempted too much, with the result that portions of the information conveyed are scrappy, while some of it is probably slightly imaginative: books and articles written in railway trains often enough provide

hard work for the reader. In a preface to his work Mr. Buckland takes pains to point out how greatly we are deficient in exact knowledge of the habits of our sea-fish, of the times and places of their spawning, of the food they eat, and of the period at which they are able to repeat the story of their birth. Some of the many questions which are asked by Mr. Buckland we are under the impression he should himself have been well able to answer. Whether cods' eggs "sink or swim" has been often discussed, and the author ought to have been able to tell us the truth in that matter; but, on turning to the account given of the cod-fish in the present book (p. 50), it seems to be singularly deficient in its details of the natural history of that animal. So far as we can observe, no reference whatever is made to the theory of Sars with reference to the floating of the eggs, but a few pages relative to the personal adventures of the author are not wanting, whilst the old story of "the Logan fish-pond” is re-told with great circumstantiality. Twenty-five pages of the work are devoted to the salmon (Salmo salar),

and the essay, confused as it is, is well worthy of perusal, although it contains, as do other portions of the book, a good deal about Mr. Buckland, and recapitulates, as usual from Land and Water, an account of some of the big fish in "my museum." It would be a tedious process to anatomise the contents of this "Natural History of British Fishes"; taking all that is written at its true value, we set down the work as an interesting collection of miscellanea. The account given of the Loch Leven trout (Salmo Levenenses) is exceedingly meagre, as is likewise the descriptions of several other fresh-water fishes, notably the vendace of Loch Maben. The most suggestive part of the present work is that which is devoted to "Pisciculture” (pp. 334 to 375). Under the title of "The Cultivation of Fish Ponds," much interesting matter is given, and a good deal of information that must be new to the uninitiated is set forth. But notwithstanding the many pleas for pisciculture which have at various times been advanced, it is questionable if the cultivation of other fresh-water fish than the salmon would pay as a food resource. A larger supply of trout would no doubt be welcome to the angler, because the trout is the fish of the angler par excellence; moreover in many places angling has now to be paid for, and lairds in Scotland who let their moors and lochs can always lease them to greater advantage when they are well stocked.

OUR BOOK SHELF

Proceedings of the Aberdeenshire Agricultural Associa tion. (Fourth Annual Report, 1879-80.) WE have here an account of the field and laboratory experiments carried out by Mr. Jamieson for the Aberdeenshire Association during the year 1879. The crops As before, experimented on were turnips and oats. parative manuring value of various phosphates in the principal object in view was to ascertain the comdifferent states of aggregation. We can glance at only a few points in the results.

Mr. Jamieson claims to have shown that a finely powdered mineral phosphate, as, for instance, powdered coprolite, is nearly equal as a manure for turnips to the superphosphate, while the simply powdered phosphate same amount of phosphate applied in a soluble form as a is of course much cheaper than the manufactured manure. There is probably no doubt that on some soils a finely powdered mineral phosphate is sufficiently soluble to produce a considerable effect on the crop, if only the phosphate is applied in sufficient quantity, so as to present a considerable surface for attack; and to Mr. Jamieson belongs the credit of giving prominence to this fact, though it was by no means unknown before his experiments. There is however no reason for supposing that dissolved and undissolved phosphates have the same manurial value. When large doses of each are applied the manures may appear of equal value, because while the undissolved phosphate is sufficient for the wants of the crop, the dissolved phosphate is in excess of all requirements, and is therefore wastefully employed. Mr. Jamieson applies 100 lbs. of phosphoric acid per acre both as dissolved and undissolved phosphate; that is to say, about 3 cwts. of bone ash and 5 cwts. of bone-ash superphosphate. Such a comparison is probably quite unfair to the soluble phosphate. For the small turnip crops obtained in Mr. Jamieson's experiments 24 cwts. of

On page 15 of the appendix the amount of phosphoric acid applied per acre is stated to be 100 lbs., but on page 16 the quantity is given as 200 lbs.

superphosphate drilled with the seed would be found quite sufficient, and probably fully equal in effect to twice the quantity of phosphoric acid applied as powdered coprolite.

Phosphate of iron applied alone was found to have practically no effect on the turnip crop, and the effect of phosphate of aluminium was but little; this is pretty much as we should expect. There is apparently some mistake in the printed analysis of the phosphate of aluminium used, as it is made to contain 38:28 per cent. of lime, and only 476 per cent. of ferric oxide and alumina.

The analyses given of the turnip soils cannot pass without a word; the reporter is surely unaware of the absurdity which these analyses present. The soil of the unmanured plot in the five experimental fields was analysed in 1876, and again in 1879, after three turnip crops had been taken. The analyses show that on an average about 20 per cent. of the nitrogen, and about 48 per cent. of the phosphoric acid in the soil had been | removed during these three years, and yet the total weight of the three turnip crops grown on the five fields during this period averaged but 16 tons per acre! The only remark made by the reporter on these figures is that the soil has evidently become reduced in nitrogen, and much reduced in phosphates; the fact that either the soil sampling or the analyses must be utterly wrong seems to have altogether escaped his attention.

The experiments with oats do not call for any special remark, except to note the patience which shelled 136,000 grains by hand in order to determine the proportion of kernel to husk in the produce of the various plots.

May we suggest that in a report of field experiments the dates of sowing and of harvest should always be given, and also a description of the character of the weather during the growing period. Without such facts before us it is impossible to interpret the results of field experiments.

Proceedings of the London Mathematical Society. Vol. xi. (November, 1879, to November, 1880).

THIS is a smaller volume than usual, there being fewer papers, and none of them of a great length. The pure mathematics prevails somewhat more than usual over the mixed.

Prof. Cayley contributes articles "On the Binomial Equation x-1=0; Trisection and Quartisection," a theorem in spherical trigonometry, on a formula of elimination. Sir James Cockle writes "On a Binomial Biordinal and the Constants of its Complete Solution." Mr. J. W. L. Glaisher, "On a Method of obtaining the 9-formula for the Sine-amplitude in Elliptic Functions"; Mr. H. W. Lloyd Tanner, "Notes on a General Method of Solving Partial Differential Equations of the First Order with several Dependent Variables," and a preliminary note on a generalisation of Pfaff's Theorem ; Mr. J. J. Walker, "Theorems in the Calculus of Operations"; and Mr. T. R. Terry, "Notes on a Class of Definite Integrals." Papers of a geometrical nature are-Mr. J. Griffiths, on a geometrical form of Landen's theorem with regard to a hyperbolic arc, and on a class of closed curves whose arcs possess the same property as two Fagnanian arcs of an ellipse; Mr. H. Hart, on the focal curves of a bicircular quartic; Mr. H. M. Taylor, on the equation of two planes which can be drawn through two given points to touch a quartic; Rev. J. Wolstenholme, a form of the equation determining the form and directions of a conic whose equation in Cartesian co-ordinates is given. Dr. Klein of Leipsic has a short note on the transformation of elliptical functions; Mr. Greenhill applies elliptic co-ordinates and Lagrange's equations of motion to Euler's problem of two centres of force; and Mr. Routh writes on functions analogous to Laplace's functions. Lord Rayleigh's papers are on reflection of vibrations at the confines of

=

two media between which the transition is gradual, and on the stability or instability of certain fluid motions. Mr. Samuel Roberts has two notes: one on a problem of Fibonacci's, and the other on the integral solution of r2 2 Py2: 22 or 22 in certain cases; Mr. R. F. Scott writes on cubic determinants and other determinants of higher class, and on determinants of alternate numbers (a treatment which he has adopted in his work Determinants"). Mr. Hugh McColl contributes a fourth paper on the calculus of equivalent statements (cf. Prof. Jevons's remarks, NATURE, vol. xxiii. p. 485). Other minor articles conclude the volume.

on

LETTERS TO THE EDITOR

[The Editor does not hold himself responsible for opinions expressed by his correspondents. Neither can he undertake to return, or to correspond with the writers of, rejected manuscripts. No notice is taken of anonymous communications. [The Editor urgently requests correspondents to keep their letters as short as possible. The pressure on his space is so great that it is impossible otherwise to ensure the appearance even of communications containing interesting and novel facts.]

The New Museum of Natural History

THE new Natural History Museum, opened on Easter Monday, was visited by some 16,000 people of a most orderly and respectable class. Owing to the great exertions of Dr. Wood, ward, whose zeal is beyond praise, the main gallery, the Pavilion, and the Gallery of Reptiles were shown in a practically completed state. The Mineral Gallery has long been ready, but the arrangement of the botanical section is still incomplete, and it was entirely closed. Some little trouble was caused with the umbrellas, and it might be worth while to consider whether, except perhaps in wet weather, the umbrellas need be taken away. The idea that people poke with sticks at objects in museums has been long exploded, and no inconvenience is felt at the Kensington Museum, the Louvre, and nearly all foreign galleries and exhibitions, where umbrellas are admitted.

The architecture in the Mammalia Gallery is very obtrusive, and its over-ornate character and the variety of tone of the terra-cotta, and the similarity of this in colour to the skulls and skeletons of the fossil mammalia, are most unfortunate.

It seems a pity that some style with more repose than "Decorated Norman" was not elected. Although very beautiful as a building, and with many features de erving high praise from an architectural point of view, it is evidently not the style best adapted to set off natural-history specimens. The cathedral-like Index Museum, with its rather dark side-chapels, and the Museum of British Zoology are of proportions that will render it difficult to make an effective display in them.

I hope that it is not finally decided to place the recent mammalia on the first floor and the birds on the ground floor, because the architect's string courses would be interfered with otherwise by the cases. The living and extinct mammalia should face each other, and the birds go aloft. Convenience time the first floor is visited the length of the Index Museum, has already been too much sacrificed to architecture. Every 150 feet, must be traversed to reach the stairs, and the same distance back along the corridor to reach the door of the Mineral Gallery. This means an immense waste of time. I also notice that the crane is close to the main entrance, and that there are no proper lifts.

If it was necessary to fashion all the ornaments from naturalhistory objects, it is a pity that the restorations were not accurately made. The oft-repeated figure of a Dapedius swallowing a fish almost its own size, and of spiral shells bent to

accommodate them to the mouldings of an arch, is not instructive. The humour of ornamenting (?) the arch leading into the pavilion with a hideously-represented Archæopteryx in high relief, repeated a dozen times, is not obvious, but some joke must doubtless be intended.

The cost of the small bronze and glass conservatories in the botanical department is out of all proportion to the objects they are to contain. Dried stems of tree-ferns and palms, though very interesting in their way, do very well in other museums without glass cases, and can be replenished for next to nothing. F. G. S.

The Tide-Predicter

MR. EDWARD ROBERTS' letter in NATURE for April 14 contains statements giving an erroneous view of the origin of the tide predicter. Any one who feels sufficient interest in the subject to derive full information will find it in my paper on "The Tide-Gauge, Tidal Harmonic Analyser, and Tide-Predicter," read before the Institution of Civil Engineers on March 1 and in the abstract of the discussion which followed it, to be published in the Minutes of the Proceedings of the Institution (vol. lxv. sess. 1880 81, part iii.), and he will see that my letter in NATURE of March 31 is correct.

The University, Glasgow, April 16

WILLIAM THOMSON

Geological Relations of Gold in Nova Scotia

IN the notice of the report of Mr. Murray on the gold of Newfoundland (NATURE, vol. xxiii. p. 472) I observe a reference to my own opinion of the age of the gold of Nova Scotia which needs so ne correction. In the second edition of "Acadian Geology" (1868) the gold-bearing series is included in the Lower Silurian, but this referred to the larger sense of that term in which it was used to include the Cambrian as well. In the third edition (1878, Supplement, pp. 81, 85, 92) I have referred this formation, on the evidence of fossils and stratigraphical position, to the age of the Lower Cambrian or Longmynd series, thus placing it on a lower horizon than the fossiliferous Primordial of Eastern Newfoundland, which I suppose to be of the age of the Acadian or Menevian group. There is therefore little difference between Mr. Murray's estimate of the age of the gold-bearing rocks of Newfoundland and my own of that of the similar rocks in Nova Scotia, except that I presume he would classify the Newfoundland series as Upper Huronian rather than Lower Cambrian. With reference to this I have been disposed to regard Mr. Murray's Aspidella slates and the associated rocks as equivalents of the Kewenian or "Upper copper-bearing group "of the West, and probably Upper Huronian, in which case they might be a little below my Nova Scotia Lower Cambrian; but the precise age of both series is deter mined merely by the fact that they appear to belong to the period between the Huronian proper, or Lower Huronian, and the Acadian group, or Menevian (Etage C. of Barrande).

It is proper to add that in the third edition of "Acadian Geology" I have shown that the filling of the Nova Scotia gold veins is much more recent than the containing rocks, and belongs to the time intervening between the Upper Silurian and the Lower Carboniferous, the richer deposits als appearing to be related to the occurrence of intrusive granites of Devonian age. There is no reason, therefore, other than the mineral character of the containing beds, why such veins might not occur in any rocks older than the Devonian, and gold discoveries have been reported in localities where the rocks are supposed to be Huronian and Silurian; but I have had no opportunity of personally verifying these statements. Thus far the important gold veins are known only in that great series of slates and quartzites of the Atlantic coast which I have referred to the Lower Cambrian. J. W. DAWSON

McGill College, Montreal, April 4

Symbolical Logic

PROF. JEVONS, in his criticism of my method in NATURE, vol. xxiii. p. 485, has stated the main points at issue between us so fully and clearly, and on the whole so fairly, that I need only say a very few words in reply.

As to the charge that my method is ante-Boolian or antiBoolian, I do not seek to repel it; on the contrary, I maintain that my method is different from Boole's in principle, and very different indeed in its practical working. The really important questions to be settled are these:

1. Are the definitions which I give of my symbols clear and unambiguous?

2. Are the rules and formula which I derive from these definitions correct?

3. Are the innovations which I propose of any practical utility?

Now, I do not think that any one who has read my papers in the Proceedings of the London Mathematical Society and my articles in Mind and in the Philosophical Magazine will refuse to answer Yes to questions 1 and 2; and with regard to question

3 I can only say that any one who answers No is bound in fairness to prove the inutility of my innovations by solving one or two of my hardest problems without their aid, and in an equally clear and concise manner. My proposal of an amicable contest in the Educational Times meant nothing more serious than this. Some of my critics (not including Prof. Jevons however) seem anxious to magnify the points of resemblance between my method and its predecessors, especially Boole's, and to minimise the points of difference. It may be as well therefore to state briefly what characteristics distinguish my method, so far as I know, from all the methods which have preceded it, and what advantages, in my opinion, accompany these characteristics.

In the first place, then, every single letter in my notation, as well as every combination of letters, denotes a statement. By this simble device I gain the important advantages of generality of expression and uniformity of interpretation and treatment. It enables me to express many important logical laws in simple and symmetrical formulæ, as, for instance,

(A : a) (B : b) (C : c) : (A + B + C: a + b + c), which otherwise could not be so expressed. To secure these advantages I sacrifice ab olutely nothing. The relations of classes, including the ordinary syllogisms, I express by speaking throughout of one individual, just as mathematicians express the properties of curves, surfaces, and volumes, by speaking throughout of the varying distances of one representative point.

He

My claim to priority on this head has been called in question on the ground that Boole too, in his equations about "secondary propositions," denotes state nents by single letters. The plain truth however is that Boole takes so ne pains to prevent his readers from imagining that he does anything of the kind. says distinctly, and in perfect consistency with the whole tenor of his book, in which he de cribes his algebra of logic as a mere offshoot and part of the ordinary algebra of quantity, that in his equations any single letter, such as x, denotes the portion of time during which some proposition x is true, the whole universe of time to which the discourse refers being the unit (see "Laws of Thought," from p. 164 to p. 170). Neither will one find any. where in Boole's work the idea (suggested to me by analytical geometry) of investigating the relations of different classes, while speaking only of one individual, and thus dispensing entirely with the quantitative words all, some, and none, which are so characteristic of the old logic.

Another peculiarity of my method is that my symbol of denil (an accent) is made repeatedly to apply to expressions of varying complexity, as, for instance, (xy)', (x + y z)'. (x : y')', leading to rules and formulae of operations, to which I find no parallel in any prior symbolic system with which I am acquainted.

Boole uses as an abbreviation for 1 - x. Let those who insist that Boole's horizontal stroke is exactly equivalent to my accent express in his notation the complex equation

(x = y) = (xy) + (y: x)',

and explain its meaning clearly without departing from Boole's quantitative interpretation of his symbols.

Lastly, my symbol: expresses implication or inference, and does not, therefore, exact'y coincide in meaning with Prof. Peirce's symbol of inclusion, as defined by him in his "Logic of Relatives," published in 1870. This symbol of inclusion, as I understand Prof. Peirce's definition of it, is simply equivalent to the words "is not greater than," and is therefore restricted to number and quantity. It is true that Prof. Peirce in his recent memoir on the "Algebra of Logic" extends the meaning of this symbol of inclusion, so as to make it also convey the same meaning as my symbol of implication; but as this memoir was published subsequently to my second and third papers in the Proceedings of the Mathematical Society, to which Prof. Peirce explicitly refers in his memoir and accompanying circular note, this later definition does not bear upon the point in discussion.

Prof. Jevons objects to my a: B as an abbreviation for a = a B, because he thinks it obscures the real nature of the reasoning operation. But one might with equal justice object on the same grounds to a3 as an abbreviation for a a a, or to the left side of the equation in the binomial theorem as an abbreviation for the right side. The symbol a: B is the exact equivalent of a = a a B, just as a = 8 is the exact equivalent of (a: B) (B: a), and I do not see that I create any obscurity by adopting in any investigation, and at any stage of the investigation, whatever form seems most suitable for the immediate purpose in view. But whether I am right or wrong in this opinion can only