Popular Science Monthly/Volume 10/March 1877/Correspondence



To the Editor of the Popular Science Monthly.

I AM a great admirer of Herbert Spencer, and especially of his wonderful "Answers to Criticisms" in your journal. When he seems entirely caught and inwoven by his adversaries, with one blow of his trenchant blade he cuts the net, and is free.

He is one of the highest of living authorities, and I read with deep attention his two editions of "The Classification of the Sciences," being particularly interested in Table I., "The Abstract Sciences." All of it but two divisions he devotes to mathematics as exactly equivalent to quantitative relations; still, at the present day, it seems an untenable cramping of mathematics to define it as the science of quantity.

A candid note in Mr. Spencer's first edition shows that it was not till after he had actually drawn up this table that he became aware of one of the most important points in the question to be solved.

It is a note to his first great division of mathematics, and says: "I was ignorant of the existence of this as a separate division of mathematics, until it was described to me by Mr. Hirst, whom I have also to thank for pointing out the omission of the subdivision 'Kinematics.' It was only when seeking to affiliate and define 'Descriptive Geometry' that I reached the conclusion that there is a negatively-quantitative mathematics as well as a positively-quantitative mathematics."

All this confession is omitted in the second edition, where, however, the much superior expression "Geometry of Position" is substituted in the table for "Descriptive Geometry," which latter was very apt to be misleading, especially to engineers, from its technical sense, in which sense, of course, Spencer did not mean it.

Now let us try to explain, in few words, what the problem was that Hirst so unexpectedly put before Spencer's mind, that you may judge whether "seeking to affiliate" it to a scheme already drawn up was a proper mental condition in which to deal with a question so important, so subtile, so profound.

Geometry, as the abstract science of space, naturally resolves itself into two great divisions, geometry of measurement and geometry of position—geometry quantitative or metrical, and geometry morphological or positional.

As an example of the first, we may take the most ordinary illustration, that of equivalent triangles. Any two triangles having the same base, and their vertices in a line parallel to that base, will be of equal or "equivalent" superficial magnitude. Although the sum of the three sides of the one triangle might be a thousand times as great as the sum of the three sides of the other, they will contain the same number of square inches or square feet. This is a metrical or quantitative proposition; but, on the other hand, many propositions are known which are purely descriptive or morphological. Take the one, perhaps, best known, the celebrated hexagram.

In any circle join any six points of the circumference by consecutive straight lines in any order: the intersections of the three pairs of opposite sides are in a straight line. Or, take any two straight lines in a plane, and draw at random other straight lines traversing in a zigzag fashion between them, so as to obtain a twisted hexagon or sort of cat's-cradle figure: if you consider the six lines so drawn symmetrically in couples, then, no matter how the points have been selected on the given lines, the three points through which these three couples of lines respectively pass will lie all in one and the same straight line. So great an authority as Prof. Sylvester has stated that this proposition "refers solely to position, and neither invokes nor involves the idea of quantity or magnitude." Take another: If any pencil of four rays is cut by a transversal, any anharmonic ratio of the four points of intersection is constant for all positions of the transversal.

Now, Carnot in his splendid "Geometry of Position," and many before and after him, have laid open a whole world of truths of this kind, truths undeniably geometrical in their nature, but founded on the primitive idea of position, and bringing in any idea of quantity only incidentally and afterward. Now, this was evidently a branch of mathematics, but, having made his scheme mathematics only coextensive with quantitative relations, Herbert Spencer must force this under the quantitative rubric, and thus was betrayed into error. Seeing that it was not really positively quantitative, he could only call it negatively quantitative, but in doing this entirely misrepresents it. In Table I. he has, under "Abstract Science:"

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"Negatively: the terms of the relations being definitely-related sets of positions in space, and the facts predicated being the absence of certain quantities (' Geometry of Position')."

Now, we contend that there is naturally nothing negative about the matter, and to call it negative is unfairly to wrest it from its proper simplicity in order to force it under a preconceived classification. The primitive and natural idea of position is of any portion of space, as distinct from space in general, and does not depend at all upon any quantitative relations, either positive or negative. But, after this, if we wish to define any position with reference to any other definite known position, we use quantities, coordinates, and by this means we can, by using only positive quantities, e. g., a positive straight line and a positive angle, accurately refer any one point in any plane to any other point in the same plane.

So "the proposition that certain three lines will meet in a point" is not "a negatively-quantitative proposition," as Spencer asserts in his note. It is primarily not quantitative at all, but positional; and, secondarily, if one wishes to look at it in a quantitative light, it is then very positively quantitative, since it asserts that the three lines will run together on a point which may be exactly fixed by positive quantities—its polar coördinates; or, having the point fixed by the intersection of either two of the lines, it asserts, directionally, that the third line must go directly through that point. In the same way, the assertion that "certain three points will always fall in a straight line" is primarily an assertion of relative position, in which the relation is defined in the simplest manner by a single positive straight line. The whole question is this: Is not position as simple and primitive an idea as quantity? and is not Spencer in error when he gives its abstract science no separate place, but ranges it under, and tries to make it depend upon, quantity?

George Bruce Halsted, A. B.,
Mathematical Fellow of Johns Hopkins University, late Mathematical Fellow of Princeton College, Intercollegiate Prizeman.

P. S.—Since the above was in print, I have noticed that Arthur Cayley holds views on this subject very much opposed to those of Mr. Spencer. (See Cayley's "Sixth Memoir on Quantics," in the "Philosophical Transactions.")G. B. H.

It has been remarked of Mr. Herbert Spencer that he does not stand well with the experts—men trained in specialties, and who know their subjects at first hand, and through and through. This is thought to be a formidable charge, and it would be formidable if it were true, and the experts agreed among themselves. But when they coincide in nothing but in differing from Mr. Spencer, we may be moderately reassured, and venture to think upon the questions they raise, without the sense of being crushed to the dust by the weight of authority.

This is not the first time that Mr. Spencer's note, or, as our contributor calls it, his "confession," has been attacked by mathematicians, and in such a way as to admonish him that, as this world is constituted, it is not always wisest to be very candid. It has ever been a rule with him carefully to acknowledge the aid he has received from others—a practice which, as in the present instance, has exposed him to misunderstanding and misrepresentation. Mr. Halsted recognizes that, by "Descriptive Geometry," Mr. Spencer did not mean those technical methods of geometrical construction to which engineers apply the name; yet no less a mathematical expert than Mr. Chauncey Wright—the pride of Cambridge, and whose biography we are soon to have—attacked hi in a dozen years ago, in the North American Review, on the very passage here dealt with by Mr. Halsted, but on the opposite ground that such was Mr. Spencer's meaning of Descriptive Geometry. And having assumed that Spencer meant a mathematical art which he was trying to classify as abstract science, Wright insinuated that by his acknowledgment to Hirst he was ignorant even of this. It was a disingenuous piece of work. Mr. Wright was then hunting through Spencer's various books in search of flaws to work up into a sensational article, and he was not very particular how he did it, so he could make a telling point. As his note was liable to such misconstruction, Mr. Spencer very naturally withdrew it in a second edition, and substituted for the title first used one less liable to be misunderstood.

And now has not Mr. Halsted also somewhat misapprehended this memorable note? If Mr. Spencer was not referring to the art of Descriptive Geometry, as Mr. Halsted admits he was not, then he must have been referring to the system of theorems in the science of pure mathematics which has grown up under the name of "Descriptive Geometry." But where is the evidence that he was ignorant of these theorems? He certainly does not say that he was made acquainted with them by Mr. Hirst, but simply that he was first informed by him that they had been grouped into "a separate division of mathematics." Why he did not know of this is readily explained, as the title Descriptive Geometry had never been adopted in England for the subject to which it had been applied, from Monge, to Reye on the Continent; and its modern restricted use was very naturally known only to professed mathematicians. What Prof. Hirst put before Mr. Spencer was, therefore, not any new mathematical problems or principles which he found it necessary as an afterthought to thrust into a previously-formed mathematical philosophy, but only the recognized differentiation of a certain mathematical province.

As for the non-quantitative mathematics, we fail to see that Mr. Halsted gets up much of a difference with Spencer. Mr. Halsted thinks that the "Geometry of Position" does not involve the notion of quantity, and Mr. Spencer thinks the same. But the experts of "Harvard" and of "Johns Hopkins" are squarely at issue on this point. After making his case against Mr. Spencer on a false interpretation of what he said, Mr. Wright admitted that, perhaps, after all, he did not mean that—possibly, instead of a branch of the engineer's art, Spencer was referring to "certain propositions in the higher geometry concerning the relations of position and direction in points and lines." But he opens a battery of sarcasms upon the idea of non-quantitative mathematics, and says of these geometrical propositions that they "cannot be made to stand alone, or independently of dimensional properties." Spencer was thus attacked by a skilled mathematician a dozen years ago for taking substantially the same ground that Mr. Halsted now advocates.

In regard to the terminology of the subject, Mr. Halsted encounters the difficulty which always arises when knowledge outgrows old definitions. No doubt, if positional geometry is non-quantitative, and is still a branch of mathematics, we should have a new definition of mathematics; but it is much easier to discredit the old one than to replace it by a better. Why does Mr. Halsted continue to apply the term geometry, which, by its very structure and etymology, implies measure and quantity, to that which has no quantity? Mr. Spencer evidently saw the difficulty; but, rather than attempt to redefine mathematical science, he preferred the alternative of marking off the newly-recognized province by a title that excluded the element of quantity—that is, he called it negatively quantitative. Mr. Halsted does not like this term. Speaking of a certain proposition given as an illustration by Spencer, he says: "It is not 'a negatively quantitative proposition,' as Spencer asserts in his note. It is, primarily, not quantitative at all." But what does Mr. Halsted suppose Mr. Spencer means by "negatively quantitative," unless he means not quantitative at all, or the denial and exclusion of quantity? Let us observe exactly whit Spencer says: "In explanation of the term 'negatively quantitative,' it will be sufficient to instance the proposition that certain three lines will meet in a point, as a negatively-quantitative proposition, since it asserts the absence of any quantity of space between their intersections. Similarly, the assertion that certain three points would always fall in a straight line is 'negatively quantitative,' since the conception of a straight line implies the negation of any lateral quantity or deviation." The italics are ours, but the statement is sufficiently explicit. The absence or negation of quantity is as strong an expression as could be used for no quantity at all, or that which Spencer calls negatively quantitative. Mr. Spencer designates the "Geometry of Position" as of this kind, and yet Mr. Halsted imputes to him the error of ranging it under and trying to make it depend upon quantity.

Mr. Halsted reports that, in his last bulletin, Cayley stands opposed to Spencer's views. It is to be* hoped that he understands him; but what is his relation to Wright and Halsted?

And now, apologizing to our readers for introducing this remote discussion, and passing it off under the head of popular science, we call upon the heirs and representatives of Johns Hopkins to hurry up their proposed Mathematical Journal, that there may be a proper place for the consideration of questions like this.



To the Editor of the Popular Science Monthly

The communication of Mr. Meehan, in your January number, and the request at its close, are herewith responded to by the entomologist in question—one to whom we may fairly apply the line—

"[Though] young in years, in sage experience old."

As the letter supplies the information called for, you will, doubtless, wish to print it in full, and I inclose it for that purpose.

Very truly yours,Asa Gray.
Cambridge, Mass., January 22, 1877.

Dear Dr. Gray: In the January number of The Popular Science Monthly, Mr. Meehan takes some exception to your note in the American Journal of Science for November in regard to the comparative abundance of insects and flowers in the Rocky Mountains of Colorado. He asks particularly for a "list of the Hymenoptera and Lepidoptera that are abundant enough in the particular part of the Rocky Mountain region covered by [his] experience, to probably act as cross-fertilizers of flowers, noting those which may perhaps be introduced since 1871." The route referred to is "through Golden City and Idaho Springs to South Park, thence to Pike's Peak and the Garden of the Gods.... to Denver over the level plateau known as the 'Divide.'" In 1873 he speaks of having visited Gray's Peak, and must, therefore, have passed up Clear Creek and through Georgetown. In 1872 I spent three months in the mountains of Colorado in company with Dr. C. C. Parry. We walked up through the canon of Clear Creek to Idaho Springs, Georgetown, and Empire City. At the latter place we established our headquarters, and there most of my collecting was done. Frequent trips were made to the neighboring mountains and canons, including the ascent of Gray's Peak. In the fall a trip was made to Middle Park. The summer of 1873 I spent in Western Wyoming with Captain Jones's exploring party. In 1874 I again visited Colorado, but spent most of my time on the plains at the base of the mountains between Boulder City and Canon City, though I made several trips into the mountains up Boulder, Left Hand, and Clear Creeks. In 1875 I spent some time in Utah among the Wahsatch Mountains. It has always been my experience that, wherever flowers were plenty, so were insects. Consequently, I have always found a botanist to be most excellent company on a collecting-trip. As my opportunities were better in 1872, my remarks refer mostly to that year, and it is not at all likely that any of the species I then noticed had been introduced. Lists of the Coleoptera, Lepidoptera, Hymenoptera, and Orthoptera, collected on this trip, have been published in the "Proceedings of the Davenport Academy of Natural Sciences," vol. i.; but I will here call attention to such of the species as seem to be most useful in the fertilization of plants:


Hymenoptera.Bombus flavifrons (Cr.) was perhaps the most common and generally distributed of the bees, though it seems to be quite a mountain species. I always found it wherever there was a patch of flowers in an opening in the timber, or at the timber-line. I did not notice that it confined itself to any particular kind of flower. It may have done so, but I do not remember it. In company with the above, though somewhat less abundant, I found B. termarius, the species mentioned by Mr. Meehan as confining its attention to Polygonum bisiorta, but I did not notice this peculiarity. Both of these species were found abundantly at Empire City and on the surrounding mountains. Besides these, Bombus borealis (Kirby), Apathus insularis, Anthophora terminalis, Megachile gentilis, Menumetha borealis, were found in abundance in the district referred to by Mr. Meehan. Of other Hymenoptera collected in this district, I mention the following, which probably were of more or less assistance in the fertilization of plants: Calliopsis (sp.?), Prosopis affinis, Agapostemen texanus, Colletes consors, Vespa diabolica, Ammophila luctuosa, A. communis, etc., besides a considerable number of smaller species as yet undetermined. For a more complete enumeration, I must refer you to the list above mentioned. To show that I am not the only one who has noticed an abundance of Hymenoptera in Colorado, I would call your attention to the papers of Mr. E. T. Cressen in the "Proceedings of the Entomological Society of Philadelphia," and particularly to a "Catalogue of Hymenoptera from Colorado Territory," published in vol. iv. of those "Proceedings."


Lepidoptera.—In the list referred to I have enumerated forty-seven species of butterflies, which I collected, with but one exception, in the mountains. I have never anywhere seen butterflies so abundant as they were in the valley of Clear Creek, between Golden City and Idaho Springs, on July 1, 1872. The air seemed literally to swarm with them. I cannot imagine how the entomologists of Mr. Meehan's party found them so scarce. Wherever there were flowers, I was sure to find butterflies, though, of course, they showed a preference to some kinds. Of the Heterocera I brought home over sixty species, mostly undetermined; but this is no indication of the actual number occurring, for I took no pains to hunt them, and only preserved what came to me.

The common morning lined sphinx (Deilephila lineata) was frequently seen at dusk, hovering about various flowers, being especially fond of the yellow thistles. I do not now recall any peculiarity regarding the other species, except that they were quite plenty. Perhaps 1872 was an unusually favorable season; but Mr. Theodore L. Mead writes that in 1871 he spent four months in Colorado, mostly in the South Park region, where he collected over 100 species and 3,000 specimens of butterflies, and 4,000 specimens of beetles, etc. I believe Mr. Mead has published an account of his observations on Colorado butterflies in the zoölogical report of Lieutenant Wheeler's explorations west of the 100th meridian. I would also refer you to an article on Coloradian butterflies, by Tryon Reakirt, in the "Proceedings of the Entomological Society of Philadelphia," vol. vi., 1866, and to the more recent works of W. H. Edwards, and others.

Although Mr. Meehan does not mention them, I have an idea that the Coleoptera and Hemiptera are often quite active agents in the fertilization of plants. Certainly the number of species of these orders found in flowers was very great, and it is more than likely that in 'going from flower to flower they carry some of the pollen with them. The Meloidæ, Chrysomelidæ, Cerambycidæ, Cleridæ, Malachidæ, Mordellidæ, etc., were especially noticeable by the large number of species and individuals. Trichodes ornatus (Say) was exceedingly abundant in the flowers of Potentilla fissa, and, after that had generally gone out of flower, on the flowers of the white and red geraniums and other plants. Owing to the fact that at the time I made these collections I knew the names of neither the plants nor insects, I cannot now remark more definitely on their habits. A full list of the species collected will be found in the "Proceedings of the Davenport Academy," vol. i.

I think what I have said shows that there is no unusual scarcity of insects in the Rocky Mountains of Colorado, at least wherever there are flowers. It should not be overlooked, however, that within the Rocky Mountain regions there are arid districts where neither insects nor flowers are particularly abundant; and also that a similar state of affairs exists in a dense pine or spruce forest. Wherever flowers are plenty in the Rocky Mountains, so are insects always; but the reverse is often not true, for I have frequently known certain insects to be exceedingly plentiful where there were no flowers. The "entomologista" of Mr. Meehan's party were certainly very unfortunate in finding so few insects. I believe Mr. Morrison, of Cambridge, an excellent collector, intends spending next summer collecting the insects of Colorado, and he will be able to add his testimony to the case.

Mr. Meehan has certainly read Lieutenant Carpenter's paper in Hayden's Report for 1873 very carelessly, or he would have seen that the five species of butterflies he speaks of as being the "doings of a whole season" were all Alpine, and collected above the timber-line, a region which a little further on he rules out of the discussion. It is certainly true that these Alpine "Lepidoptera are undoubtedly peculiar to high latitudes and great elevations;" but the species found lower down in the cañons often show a greater affinity to Mexican and Californian types. I was more fortunate than Lieutenant Carpenter, and took over twenty species of butterflies above the timber-line.

I have endeavored to show that sometimes at least insects are quite plentiful in the Colorado mountains. They are certainly more plentiful in the mountain-regions than on the plains.

Yours very truly,
J. Duncan Putnam.
Davenport, Iowa, January 10, 1877.


To the Editor of the Popular Science Monthly.

The Superintendent of the Ninth Census, while showing the causes of loss in population produced by the late war, neglected to point out the actual decrease incident to this cause, as shown by the State censuses of 1865.

The retarding influence may be seen by reference to the States of New York and Massachusetts.

In the former State, whereas the regular increase for each period of five years had been nearly 500,000, from 1860 to 1865 there was a decrease of 29,000. From 1865 to 1870 there was an increase of 529,000.—("Manual" for 1870.)

In Massachusetts the increase of population from 1850 to 1855 was 138,000; from 1855 to 1860 it was 90,000; from 1860 to 1865, 36,000; from 1865 to 1870, 190,000; and from 1870 to 1875, 194,000. The regularity of increase in the State is shown by the fact that the difference between the actual population in 1870 and that computed on the supposition that the increase was in arithmetical progression was only 2,120.—("Massachusetts State Census" for 1875, vol. i., p. xxxii.)

In these two States, therefore, the increase of population during the war-period was only 7,000, while in the next five-year period it was fully a hundred times as great. The population of these two States was over 6,000,000, or more than fifteen per cent, of the total population of the United States.

As it may be urged that these States suffered heavily in loss of immigration, we will attempt to estimate their actual loss in this respect. The total loss of immigration is estimated ("Ninth Census," vol. i., p. xix.) at 350,000. If we suppose the immigration into a State to be proportionate to the foreign population of that State, the proportion of loss in immigration for these two States will be 27 per cent. of the whole loss, or about 94,500. Omitting the loss in immigration, therefore, the total gain of these two States will be 101,500. Even if we further suppose that these States suffered another special loss of 48,500, the total gain would still be only 150,000. Multiplying this by 8, the ratio between the increase of population in these States (700,000) from 1865 to 1870, and the estimated increase of the United States for the same time, we get for the total gain of the country, without considering the loss in immigration, 1,200,000. Deducting this loss, we have for the entire gain 850,000.

We have no reason to suppose that New York and Massachusetts were especial sufferers during this period. Many of the Southern States probably suffered more, especially in loss of negro population, which amounted to half a million during this period ("Ninth Census," vol. i., p. xviii.). There is, then, no reason to suppose that the above estimate falls far short of the truth. Even if the estimate is increased to 1,500,000, which seems improbable, the population of the country in 1865 would still fall short of 33,000,000. This would indicate for the succeeding period of five years an increase of 5,500,000, which is somewhat above the average. That this large increase is actual is rendered probable by the corresponding large increase in Massachusetts and New York for the same time.

Alexander Duane, Union College.