a+b=b+a

a + (b + c) = (a + b) + c

b c = c b

(ab) ca (bc)

a (m + n) = am + an

we have a series of identities whose truth or falsity is entirely undeterminable. In order, therefore, fully to define those operations, we will say that all propositions, equations, and identities which are in the general case left by the former definitions undetermined as to truth shall be true, provided they are so in all interpretable cases.

On Arithmetic.

Equality is a relation of which identity is a species.

If we were to leave equality without further defining it, then by the last scholium all the formal rules of arithmetic would follow from it. And this completes the central design of this paper, as far as arithmetic is concerned.

Still it may be well to consider the matter a little further. Imagine, then, a particular case under Boole's calculus, in which the letters are no longer terms of first intention, but terms of second intention, and that of a special kind. Genus, species, difference, property, and accident, are the well-known terms of second intention. These relate particularly to the comprehension of first intentions; that is, they refer to different sorts of predication. Genus and species, however, have at least a secondary reference to the extension of first intentions. Now let the letters, in the particular application of Boole's calculus now supposed, be terms of second intention which relate exclusively to the extension of first intentions. Let the differences of the characters of things and events be disregarded, and let the letters signify only the differences of classes as wider or narrower. In other words, the only logical comprehension which the letters considered as terms will have is the greater or less divisibility of the classes. Thus, n in another case of Boole's calculus might, for example, denote "New England State"; but in the case now supposed, all the characters which make these States what they are being neglected, it would signify only what

essentially belongs to a class which has the same relations to higher and lower classes which the class of New England States has, - that is, a collection of six.

In this case, the sign of identity will receive a special meaning. For, if m denotes what essentially belongs to a class of the rank of "sides of a cube," then mn will imply, not that every New England State is a side of a cube, and conversely, but that whatever essentially belongs to a class of the numerical rank of "New England States" essentially belongs to a class of the rank of "sides of a cube, and conversely. Identity of this particular sort may be termed equality, and be denoted by the sign=.* Moreover, since the numerical rank of a logical sum depends on the identity or diversity (in first intention) of the integrant parts, and since the numerical rank of a logical product depends on the identity or diversity (in first intention) of parts of the factors, logical addition and multiplication can have no place in this system. Arithmetical addition and multiplication, however, will not be destroyed. a b c will imply that whatever essentially belongs at once to a class of the rank of a, and to another independent class of the rank of b belongs essentially to a class of the rank of c, and conversely. a+b=c implies that whatever belongs essentially to a class which is the logical sum of two mutually exclusive classes of the ranks of a and b belongs essentially to a class of the rank of c, and conversely. It is plain that from these definitions the same theorems follow as from those given above. Zero and unity will, as before, denote the classes which have respectively no extension and no comprehension; only the comprehension here spoken of is, of course, that comprehension which alone belongs to letters in the system now considered, that is, this or that degree of divisibility; and therefore unity will be what belongs essentially to a class of any rank independent of its divisibility. These two classes alone are common to the two systems, because the first intentions of these alone determine, and are determined by, their second intentions. Finally, the laws of the Boolian

*.Thus, in one point of view, identity is a species of equality, and, in another, the reverse is the case. This is because the Being of the copula may be considered on the one hand (with De Morgan) as a special description of "inconvertible, transitive relation," while, on the other hand, all relation may be considered as a special determination of being. If a Hegelian should be disposed to see a contradiction here, an accurate analysis of the matter will show him that it is only a verbal one.

calculus, in its ordinary form, are identical with those of this other so far as the latter apply to zero and unity, because every class, in its first intention, is either without any extension (that is, is nothing), or belongs essentially to that rank to which every class belongs, whether divisible or not.

These considerations, together with those advanced on page 293 (§ 12) of this volume, will, I hope, put the relations of logic and arithmetic in a somewhat clearer light than heretofore.

Five hundred and eighty-sixth Meeting.

October 8, 1867.- MONTHLY MEETING.

The CORRESPONDING SECRETARY in the chair.

The Corresponding Secretary read letters relative to exchanges; also a letter from Major-General Sabine in acknowledgment of his election as Foreign Honorary Member of the Academy.

The Corresponding Secretary announced the recent decease of Hon. Charles G. Loring, of the Resident Fellows.

Dr. C. G. Putnam presented the meteorological observations of the late Dr. Jackson.

Professor Lovering presented for Professor Treadwell the following paper:

Corrections to a Paper "On the Comparative Strength of Cannon of Modern Construction," published in Vol. VII. of the Proceedings of the Academy. By DANIEL TREADWELL.

IN a paper "On the Comparative Strength of Cannon of Modern Construction," written by me in January, 1866, communicated to the Academy in September of the same year, and published in the last volume (the seventh) of our Proceedings, I, by some inadvertence for which I am now unable to account, in computing the force of the 600 pounder, or 13.3-inch coil gun, as constructed by Armstrong, described it as capable of bearing a charge of 100 pounds of powder.

Although this quantity of powder was no doubt fired in it, I know not how many times, yet it ought not by any means to be rated as its service charge; and I recognize it as an oversight in me to have taken

it as such. In fact, I have no belief that more than 70 pounds of powder should be assigned as the service charge of the Armstrong 13.3-inch gun; as no gun can be trusted for long-continued firing with more than 7% of the largest charge of powder which it may have withstood, and no cast-iron gun with so much as this.

I have not the data necessary to determine accurately the velocity, and consequently the force, which this reduction of the charge of powder must make in the shot; but if we take the force of the shot in the direct ratio of the weight of the charge of powder, we shall have 261, instead of 372.8, as representing the "number of pounds of shot raised one foot by each pound of metal in the gun," as these numbers are in the ratio of 70 to 100.

I am not able to state what has constituted the greatest charge of powder borne by Armstrong's gun of 12 tons, carrying a shot of 300 pounds; but reducing the charge of 60 pounds, as given by me in the ratio of 70 to 100, we have a charge of 42 instead of 60 pounds of powder, and a consequent reduction of the force of the gun from 392 "foot pounds" to 273 "foot pounds."

I have thought it the more necessary to make this correction, as in a computation of the force of the Dahlgren and Rodman guns, given in the same paper to which this is a correction, the quantity of powder then understood by me from all that had been published by government authority as constituting a service charge was taken as one of the factors in assigning the measure of the force to those guns. It is now claimed, however, to have been discovered that the Rodman gun is capable of withstanding much larger charges of powder than were authorized to be used when my paper was communicated to the Academy.

Professor Lovering made the following remarks on the Optical Method of studying Sound, illustrating the subject by many experiments:

When the science of Acoustics is studied by means of the ear exclusively, we judge of the process simply by the result, that is, by the sensation. The optical method of investigation often gives us an insight into the process itself. Sound begins with a stationary vibration in the sonorous body; it is propagated by a progressive undulation; and it ends, physically and mechanically considered, in a vibration of some one of the three thousand nervous filaments discovered by Corti in

the labyrinth of the human ear. Whether we regard the sound, therefore, at its origin, in its promulgation, or in the sensation, it is nothing but a vibration; and vibration is motion, and motion is the subject of vision. So that to see sound is only to see the motions which cause it. The only difficulty in seeing sound lies in the fact that the acoustic vibrations are upon a microscopic scale of magnitude, and, by their quick succession, the separate effects of individual vibrations blend into one sensation, in the eye as well as in the ear, by virtue of what is called in both cases the persistency of the impression - on the organ of sensation. To overcome the first difficulty a beam of light is reflected from the vibrating body, or a mirror attached to it, which moves in angle twice as fast as the body itself, while the motion in arc may be amplified to any extent by increasing the length of the beam of light. The second difficulty is surmounted by reflecting the vibrations of the sonorous body itself, or some more visible effect which they originate, from a revolving mirror. By this device of looking at the image of the body, instead of the body itself, its vibrations, which coexist in space, are disentangled from each other, and individual vibrations, hundreds of which succeed each other in a single second of time, are translated into a long belt of space, in which even two successive ones do not overlap.

The optical method of studying sound embraces, in general, Savart's contrivance for discovering and exhibiting the nodal lines of plates by means of sand sprinkled over their surface, the investigation of the nodes and bellies of sounding strings by mounted riders, and of columns of air by a little drumhead suspended in the pipes, and, more recently, Lissajous's mirrors attached to tuning-forks, etc., Koenig's flames played upon by vibrating columns of air and reflected in a revolving mirror, and, finally, Melde's strings excited by the sympathetic vibration of an attached tuning-fork or bell.

The present communication is confined, however, to Koenig's reflected flames, in which are seen the individual vibrations of an organpipe; by which can be beautifully demonstrated to the eye: First, That the number of vibrations increases with the audible pitch; Second, - That coexisting vibrations produce maxima and minima of motion corresponding to the beats which are recognized by the ear; Third, - That one column of air will respond, in sympathetic vibration, to another, when there is an agreement between their fundamental notes or some of their harmonies; Fourth, That two unison-pipes, brought