NOTICES AND ABSTRACTS OF MISCELLANEOUS COMMUNICATIONS TO THE SECTIONS. MATHEMATICS AND PHYSICS. MATHEMATICS. On Plane Stigmatics. By ALEXANDER J. ELLIS, F.R.S. THIS paper is a continuation of that read at the Bath Meeting (Report of the Brit. Assoc. 1864, Trans. of Sections, p. 2). By means of diagrams the meaning of the stigmatic line, stigmatic involution, stigmatic homography, and the stigmatic circle was illustrated, and these were shown to include the straight line and circle of Descartes, and the involution and homography of Chasles, as particular cases. The imaginary points of intersection of a stright line and circle, the imaginary double points, and imaginary double rays of an involution and homography, &c. were, for the first time, publicly exhibited on paper. And an attempt was thus made to show that the principle of stigmatics, as explained in the papers cited, affords a complete solution of the problem of the geometrical signification of imaginaries in the geometries of Descartes and Chasles, and establishes an unbroken agreement between ordinary algebra and plane geometry. On Practical Hypsometry. By ALEXANDER J. ELLIS, F.R.S. If the heights of the barometer be B, b inches, the temperatures of the air A, a degrees Fahr., and the temperatures of the mercury M, m degrees Fahr. at the lower and upper stations respectively, then, for all British heights, the difference of the level of the two stations is [B-b)×52400 −2}. (M−m) |×: 900 English feet to the nearest unit, with the same accuracy as by Laplace's complete formula. Beyond the British isles, small corrections for the alteration of gravity in latitude and on the vertical, have to be made, but when aneroid barometers are used, these corrections may be neglected, as being much inferior to the probable instrumental error. The heights must be taken in sections not exceeding 4000 feet each, both on account of the construction of the formula, and of the unknown law of variation of temperature. When the decrease of the temperature of the air does not vary nearly as the decrease of the height of the barometer, or the observations at both stations, when distant, are not simultaneous, or the two stations are not nearly north-east and south-west of each other, the results of barometric hypsometry must be received with great caution. 1866. 1 Description of a New Proportion-Table, equivalent to a Sliding-rule 13 feet 4 inches long. By J. D. EVERETT, D.C.L., Assistant Professor of Mathematics in Glasgow University. The distinctive feature of the new arrangement consists in breaking up each of the two pieces which compose a sliding-rule into a number of equal parts, and arranging these consecutively in parallel columns, the columns in one of the two pieces being visible through openings cut between the columns of the other. The largeness of the scale is such that the space from 1 to 1·1 is divided into a hundred parts, the smallest of these being of an inch long. The material employed is Bristol board, printed from copper plates, the dimensions of each board, exclusive of margin, being 16 by 8 inches. Multiplication and division can be performed by this Table with the same accuracy as by four-figure logarithms, and with greater ease and expedition. Formulæ not adapted to logarithmic computation are thus rendered available, and with the aid of a small table of natural sines and tangents the calculations of nautical astronomy can be performed with great facility and with all needful accuracy. On certain Errors in the received Equivalent of the Metre, &c. By F. P. FELLOWS. On Tschirnhausen's Method of Transformation of Algebraic Equations, and some of its Modern Extensions. By the Rev. Prof. R. HARLEY, F.R.S. It has long been known that any algebraic equation may be deprived of its second term by a linear transformation. Tschirnhausen introduced quadratic, and suggested higher transformations, and thus opened the way to great progress in the theory. He showed that by the solution of a linear equation and of a quadratic, any algebraic equation may be deprived of its second and third terms simultaneously. The complete quintic may in this way be reduced to a quadrinomial form. Erland Bring, Professor of History in the University of Lund, in a paper bearing date 14th December 1786, seems to have been the first to extend Tschirnhausen's method so as to reduce the quintic to a trinomial form by depriving it of its second, third, and fourth terms simultaneously. (See a paper by Prof. Harley, entitled "A Contribution to the History of the Problem of the Reduction of the General Equation of the Fifth Degree to a Trinomial Form," Quarterly Journal of Mathematics, vol. vi. 38-47.) Bring's process has lately been simplified by Mr. Samuel Bills of Hawton, near Newark; and Prof. Harley explained to the Section how Mr. Bills's method might be extended so as to deprive the general equation of the nth degree of its second, third, and fourth terms by the solution of equations none of which rise higher than the third degree; and of its second, third, and fifth terms by the solution of equations none of which rise higher than the fourth degree. (See "Mathematics from the Educational Times," vol. i. pp. 8, 38-40, 57, 58.) Notice was taken of the labours of other investigators in the same field, particularly Mr. Jerrard, Sir W. R. Hamilton, Chief Justice Cockle (Queensland), Prof. Cayley, and Prof. Sylvester. The author concluded with some observations on the alleged solutions of the general quintic by the late Mr. Jerrard and Judge Hargreave. pp. On Differential Resolvents. By the Rev. Professor R. HARLEY, F.R.S. The author gave a short account of his researches on differential resolvents, particularly those connected with certain trinomial forms of algebraic equations. An abstract of these researches has recently been published by the London Mathematical Society. He also pointed out the coincidence of some of his own results with those obtamed about the same time, quite independently, by Chief Justice Cockle, F.R.S., of Queensland. The differential equation br x dn dx u=an [n—r L N da (in which the ordinary factorial notation [•]°=• (•−1) (9—2) ... (−0+1) is adopted) is satisfied by the mth power of any root of the algebraic equation y" -axy" "+b=0, u being considered as a function of x. This theorem implicitly involves the following, which was communicatd to the author by Chief Justice Cockle in a letter under date Brisbane, Queensland, Australia, October 17-18, 1865. The differential equation for On this result Chief Justice Cockle, in the same letter, remarks, "The conditions under which (2) is immediately depressible by one or two orders are, that one or both of the relations should be satisfied; a and ß being integers comprised between the limits 0 and n-1 both inclusive (zero I treat as an integer), and p being an integer comprised between 1 and both inclusive, and σ being an integer comprised between 1 and n-r both inclusive. If both conditions are satisfied, but a=ẞ then (2) is immediately depressible by one order. If (only) one condition be satisfied, the same thing holds. If both be satisfied, and a-B does not vanish, (2) is depressible by two orders." Remarks on Boole's Mathematical Analysis of Logic. The author's remarks were arranged under three heads. First, he gave some account of Boole's system as developed in his Mathematical Analysis of Logic,' and more elaborately in his great work on 'The Laws of Thought.' Next, he noticed some remarkable anticipations of Boole's views. And in the concluding portion of his paper he pointed out the direction in which he believed Boole's method might be usefuly extended. 1. He contended that in Boole's system the fundamental laws of thought are deduced, not, as has sometimes been represented, from the science of number, but from the nature of the subject itself. Those laws are indeed expressed by the aid of algebraical symbols, but the several forms of expression are determined on other grounds than those which fix the rules of arithmetic, or more generally of algebra; they are determined in fact by a consideration of those intellectual operations which are implied in the strict use of language as an instrument of reasoning. In algebra letters of the alphabet are used to represent numbers, and signs connecting those letters represent either the fundamental operations of addition, subtraction, &c., or, as in the case of the sign of equality, a relation among the numbers themselves. In Boole's calculus of logic literal symbols (r, y, &c.) represent things as subjects of the faculty of conception, and other symbols (+, - &c.) are used to represent the operations of that faculty, the laws of the latter being the expressed laws of the operations signified. For instance, r+y stands in this system for the aggregation of the classes or collections of things represented by a and y, and xfor what remains when from the class or collection a the class or collection` y is withdrawn; xxy, or more simply ay, represents the whole of that class of things to which the names or qualities representeed by x and y are together applicable; and = expresses the identity of the classes x and y. The canonical forms of the Aristotelian syllogism are really symbolical; but the symbols are less perfect of their kind than those which are employed in this system. By adopting algebraical signs of operation, as well as literal symbols and the mathematical sign of equality, Boole was enabled to give a complete expression to the fundamental laws of reasoning, and to construct a logical method more self-consistent and comprehensive than any hitherto proposed. His calculus does not involve a reduction of the ideas of logic under the dominion of number; but it rests on a fact which its inventor -y has rigorously established, viz., "that the ultimate laws of logic-those alone upon which it is possible to construct a science of logic-are mathematical in their form and expression, although not belonging to the mathematics of quantity." The term mathematics is here used in an enlarged sense, as denoting the science of the laws and combinations of symbols, and in this view there is nothing unphilosophical in regarding logic as a branch of mathematics, instead of regarding mathematics as a branch of logic. The symbols of common algebra are subject to three laws, viz. These laws are fundamental; the science of algebra is built upon them. And they are axiomatic; each of them becomes evident in all its generality the moment we clearly apprehend a single instance. Now Boole has shown that the same laws govern the symbols of logic, and that therefore in the logical system the processes of algebra are all valid. But at the root of this system there is found to exist a law, derived from the nature of the conception of class, to which the symbols of common algebra are not in general subject. This law is named by Boole "the law of duality," and is expressed by the equations Now viewing the equation = as algebraic, the only values which will satisfy it are 0 and 1. If therefore an algebra be constructed in which the symbols x, y, z, &c. admit indifferently of the values 0 and 1, and of these alone, it follows that "the laws, the axioms, and the processes of such an algebra are identical in their whole extent with the laws, the axioms, and the processes of an algebra of logic." Difference of interpretation alone divides them. Upon this principle Boole's logical method is founded. Propositions are represented as equations; these are dealt with as algebraic, the literal symbols involved being supposed susceptible only of the values 0 and 1; all the requisite processes of solution are performed; and finally the logical interpretation of the symbols is restored to them. Some illustrations were given of the application of the method. That method, to use the originator's own words, "has for its object the determination of any element in any proposition, however complex, as a logical function of the remaining elements. Instead of confining our attention to the 'subject' and the 'predicate,' regarded as simple terms, we can take any element or any combination of elements entering into either of them, make that element or that combination the subject' of a new proposition, and determine what its 'predicate' shali be, in accordance with the data afforded to us." In the same way any system of equations whatever, by which propositions or combinations of propositions can be represented, may be analyzed, and all the "conclusion" which those propositions involve be deduced from them. 2. Bacon, in his 'Novum Organum,' Liber Secundus Aphorismorum, A. XXVII., notices incidentally an analogy that exists between a well-known axiom in mathematics, and the fundamental canon of syllogism: he says, "Postulatum mathematicum, ut quæ eidem tertio æqualia sunt etiam inter se sint æqualia, conforme est cum fabrica syllogismi in logica, qui unit ea quæ conveniunt in medio." On this passage R. Leslie Ellis remarks, "The importance of the parallel here suggested was never understood until the present time, because the language of mathematics and of logic has hitherto not been such as to permit the relation between them to be recognized. Mr. Boole's Laws of Thought' contain the first development of ideas of which the germ is to be found in Bacon and Leibnitz; to the latter of whom the fundamental principle, that in logic a2=a, was known (vide Leibnitz, Philos. Works, by Erdmann, p. 130). It is not too much to say that Mr. Boole's treatment of the subject is worthy of these great names. Other calculuses of inference (using the word in its widest sense), besides the mathematical and the logical, yet perhaps remain to be developed." (Bacon's Collected Works, vol. i. footnote on p. 281.) The reference to Leibnitz requires some correction, for on p. 130 of the edition cited, there is nothing whatever relating to the logical question. Probably the passage intended is that which occurs on p. 103, where Leibnitz, in a paper entitled "Difficultates quædam logica," makes a near approach to the enunciation of the fundamental law of logic, although he does not, either in that paper or elsewhere, so far as is known, state the law explicitly. He does, however, observe that AB=BA, which is Boole's law of commutation; and further, that from the proposition, all A is B, we may infer that AB-A,—an inference which, applied to the identical proposition A is A, gives us Boole's law of duality, AA=A. One of Leibnitz's illustrations of this inference is very curious. "Quidam se appellabat GRÜNBERG, viridis mons. Sodalis ei dicit, sufficeret ut Te appellares Berg, mons. Quid ita? respondet prior, putasne omnes montes esse virides? Cui sodalis, ita, inquit, nunc certe, nam æstas erat. Ita illi naturalis sensus dictabat hæc duo coincidere, omnis mons est viridis et æquivalent viridis mons et mons." Boole did not become aware of these anticipations by Leibnitz until more than twelve months after the publication of his 'Laws of Thought,' when they were pointed out to him by R. Leslie Ellis. Ellis subsequently addressed to Boole some "Observations," which yet remain unpublished, on some of the elementary parts of his system. These "Observations," which are chiefly critical, throw much light on the writer's views respecting the possibility of developing other calculuses of inference besides the mathematical and the logical. 3. The space at our disposal for this abstract will not permit us to print Ellis's paper here in extenso, but the following brief extract will give a tolerably clear idea of its general character. "It appears to be assumed in Chap. III. Section 8 ['Laws of Thought'], that in deriving one conception from another the mind always moves, so to speak, along the line of predicamentation, always passes from the genus to the species. No doubt everything stands in relation to something else, as the species to the genus, and consequently the symbolical language proposed is in extent perfectly general,. that is, it may be applied to all the objects in the universe. But I venture to doubt whether it can express explicitly all the relations between ideas which really exist, all the threads of connexion which lead the mind from one to the other. It seems to me that the mind passes from idea to idea in accordance with various principles of suggestion, and that in correspondence with the different classes of such principles of suggestion, we ought to recognize different branches of the general theory of inference. This leads me to a further doubt whether logic and the science of quantity can in any way be put in antithesis to one another. From the notion of an apple we may proceed to that of two apples, and so on in a process of aggregation, which is the foundation of the science of discrete quantity. Or again, from the notion of an apple we may proceed to that of a red apple, and this movement of the mind in linea predicamentali is the foundation of ordinary logic. But it is plain à priori that there are other principles of suggestion besides these two, and the following considerations lead me to think that there are other exercises of the reasoning faculty than those included in the two sciences here referred to. In the first place, certain inferences not included in the ordinary processes of conversion and syllogism were recognized as exceptional cases by the old logicians. Leibnitz has mentioned some with the remark that they do not depend on the dictum de omni et nullo, but on something of equivalent evidence. The only question is whether we should be right in considering these cases as exceptions, and if they are so, to what they owe their existence. One instance is the inversio relationis, e. g. Noah is Shem's father, therefore Shem is Noah's son. Here we pass from the idea of Shem to that of his father, and vice versa. The movement of the mind is along a track distinct from that which we follow, either in algebra or what we commonly call logic. The perception of the truth of the inference depends on a recognition of the correlation of the two ideas, father and son." The author gave his reasons for believing that, when the "exceptional cases" referred to in the above passage are fully investigated, and a calculus is devised for their symbolical solution, it will be found that the processes involved in such a calculus formally coincide with the processes commonly employed in the solution of functional equations. He also pointed out that it was in this direction probably that Boole's method would be found to admit of extension-an extension analogous to that which Boole himself effected for the theory of the solution of differential |