To Mary Somerville   [23 February 1830]

Slough. Feb[ruary] 23 1830.

My dear Madam.

I send you for Mr Richards the first 40 pages of your MS. I have here and there appended some pencil notes which you can easily rub out, and I take the opportunity here to make one or two remarks not so well inscribable in the blank pages, and of a more general nature.

As you contemplate separate publication, and as the attention of many will be found to a work from your pen for those who will just possess quantum enough of mathematical knowledge to be able to read the first chapter without being able to follow you into its applications, and as as these moreover are the very people who will think themselves privileged to criticise & use their privilege with the least discretion, I cannot recommend too much clearness, fullness & order in the exposé of the principles. Were I you I would devote to this first part at least double the space you have done. Your familiarity with the results and the formulae has led you into what is definitely natural in such a case — a somewhat hasty passing over what to a beginner would prove insuperable difficulties, and if I may so express it, a sketchiness of outline (as a painter you will understand my meaning & what is of more consequence, see how it is to be remedied).

You have adopted I see, the principle of virtual velocity, and the principle of D’Alembert, rather as separate & independent principles to be used as instruments of investigation than as convenient theories, flowing them selves from the general law of force & equilibrium, to be first proved and then remembered as compact statements in a form fit for use. The demonstration of the principle of virtual velocities is so easy & direct in Laplace that I cannot imagine anything capable of rendering it plainer than he has done. But a good deal more explanation of what is virtual velocity, &c., would be advantageous — and virtual velocities should be kept quite distinct from the arbitrary variations represented by the sign $\delta$.

With regard to the principle of D’Alembert — take my advice and explode it altogether. It is the most awkward and involved statement of a plain dynamical equation that ever puzzled students. I speak feelingly and with a sense of irritation at the whirls & vortices it used to cause in my poor head when first I entered on this subject in my days of studentship. I know not a single case where its application does not create obscurity — nay doubt. Nor can a case ever occur where any such principle is called for. The general law that the change of motion is proportional to the moving force & takes place in its direction, provided we take care always to regard the reaction of curves, surfaces, obstacles, &c., as so many real moving forces of (for a time) unknown, magnitude, will always help us out of any dynamical scrape we may get into. Laplace, page 20, Mec. Cel. art 7. is a little obscure here, and in deriving his equation (f) a page of explanation would be well bestowed.

One thing let me recommend, if you use as principles either this, or that of virtual velocities or any other, state them broadly & in general terms. See Mec. Cel. p12. “Si l’on fait varier infiniment peu &c”. Indeed a little more distinctness of in the enunciation of theorems in their detached, insulated form, ready for any applications, will be desirable throughout.

Allow me too to observe that you might take for granted that your reader understand a good deal of algebra, & geometry, &c. In Consequence, were I to advise, I should dispense with all those papers where very elementary abstract properties, and very common methods are stated. such as the passages I have marked in pages 21, 17, 19, &c.

In page 32 I confess I do not apprehend the connexion of the analysis. & in the equation

$$\{P-\frac{{\partial }^{2}x}{\partial t^2}\}\delta x+\{Q-\frac{{\partial }^{2}y}{\partial t^2}\}\delta y+\{R-\frac{{\partial }^{2}z}{\partial t^2}\}\delta z-\lambda \delta u=0$$

is not, as you make it, the reaction of the surface. That reaction is really

$$\lambda \times \sqrt{{\left(\frac{du}{dx}\right)}^2+{\left(\frac{du}{dy}\right)}^2+{\left(\frac{du}{dz}\right)}^2}$$

and your equation (13) as it stands, is assumed, & not proved. The analytical artifice by which the addition of the term $-\lambda \delta u$ added to the general equation, where $\lambda$ is an indeterminate, render it equivalent to the two $(P-\frac{{\partial }^{2}x}{\partial t^2}\delta x+\&c=0$ and $\delta u=0$ is too beautiful and too useful in all similar cases not to merit a distinct explanation, and in fact, without much circumlocution I do not see how its use is to be avoided.

You will think me, I fear, a rough critic, but I think of Horace’s good critic Fiet Aristarchus, nec dicet, cur ego amicum, Offendam in nugis? Hæ nugæ seria ducent, In mala, — and what we can both now laugh at, & you may, if you like, burn as nonsense — (I mean these remarks) — would come with a very different kind of force from some sneering reviewer in the plenitude of his triumph at the detection of a slip of the pen or one of those little inaccuracies which humana parum cavit natura.

I think you would find a regular system of numbering the paragraphs (each paragraph to be numbered as it begins on a fresh line in a page) extremely convenient for reference. I find such an aid invaluable.

Mrs Herschel desires her kind regards, & I remain dear madam

Very faithfully yours,

J. F. W. Herschel.