Before turning to consider the fundamental insights that gave rise to *The Principia*, I want to touch on the question of how we approach a treatise on Mathematical Physics that was written over 200 years ago. This is a matter that concerns the reader as well as the translator. Some translators seek to transport the reader to the time and place of the writer, and some to translate the writer into the time and place of the translator.

It makes no sense to ask for a translation of *The Principia* as Newton might have written it had he lived today. But there are problems with trying to transport the reader back to the time of Newton. We are all changed by technological advance. We have amazingly accurate watches, but one of the ways in which Newton timed the fall of balloons, dropped from a height in St. Paul’s cathedral, was to use a pendulum. We expect to be able to measure lengths to a ridiculous degree of accuracy against a universally defined standard. We are used to travelling smoothly at speed, so we realise, as if by instinct, that if we drop a sandwich in a fast train moving steadily through the countryside, that it will fall straight downwards, as perceived by the passengers in the compartment.

Similarly, we use mathematical and physical words in a way that has changed – I would say has advanced – from the way that Newton used them. So to translate Newton word for word often conceals his meaning beneath a veil that is likely not only to lose the reader, but that seems, on occasion, to lose the translator as well.

Newton looks back to Euclid, to Apollonius, to Cicero. Here is Proposition 18, Book 5, of Euclid’s *Elements*, as it appears in Todhunter’s edition.

PROPOSITION 18. THEOREM

If magnitudes, taken separately, be proportionals, they shall also be proportionals when taken jointly; That is, if the first be to the second as the third to the fourth, the first and second together shall be to the second as the third and fourth together to the fourth.

Euclid’s proof of this proposition extends to three pages. But the reader who takes the trouble to decode the proposition will see that it is trivial primary school arithmetic. This is a suitable way to translate this proposition as it is only of historical significance. But a modern reader who is not used to this cumbersome form of expression is likely to be completely thrown by a translation of one of Newton’s tricky arguments in this style. Moreover, learned readers in Newton’s time, with their classical education, were used to digesting far longer sentences than most of us can manage now. On the other hand, a translation is a translation, and not a paraphrase; so I have constantly had to compromise.

There are many other stylistic conventions that separate us from Newton.

Let us begin at the beginning. The Principia starts with a chapter on DEFINITIONES, followed by one on AXIOMATA, SIVE LEGES MOTUS. That is DEFINITIONS, and AXIOMS, OR THE LAWS OF MOTION. The great book starts with:

DEFINITIO 1

Quantitas materiae est mensura eiusdem orta ex illius densitate et magnitude conjunctim.

That is: Definition 1. The quantity of material is a measure of the same that has arisen from its density and magnitude together.

Here I have given a word for word translation, but what do we learn from this first sentence?

The first thought that comes to mind is that this is a useless definition, and Newton never refers back to it. How does he propose to define ‘density’? So I ask; what is the definition of a definition? I suggest that the answer for Newton, as with Euclid and his useless definitions of a point, a line, and a surface, is that he hopes to make the reader feel comfortable with the term defined, rather giving a proper definition.

So I ask historians: What have you to say about the development of the definitions in Physics? And I ask the physicists: How should Newton have defined mass? He is referring here to inertial mass, not to gravitational mass. Presumably he needs to define mass in the context of the laws of motion.

Another point is his use of *conjunctim*, or ‘together’. Linguistically, Newton might have meant the sum of the density and the volume, or the product of the density and the volume. From the context it has to be the product. But what if the density is not uniform? Then you need a limiting process that involves dividing the body into a large number of small bits whose density is approximately constant, calculating the mass of each bit, and adding. The limiting process consists of considering finer and finer subdivisions of the body.

This process is now called ‘integration’. For it to work the shape of the body and the density must not behave too wildly; experts will consider the Lebesgue integral. Newton uses a version of the much simpler Riemann integral, which he defines later with uncharacteristic clarity. As the body is three dimensional, this will be a triple integral, and Newton did calculate a triple integral.

But, you may ask: How do you know that Newton was not thinking of a body of constant density? Of course I don’t know, and I don’t think that Newton knew either. He frequently makes statements that can be interpreted in varying degrees of generality, and quotes them with different meanings. It is mathematically trivial to obtain the case of constant density from the case of varying density, and *vice versa*. But modern practice requires us to say precisely what we mean. So this first sentence is, to that extent, untranslatable.

I don’t want to bore the reader with enumerating all the possible meanings of what Newton wrote at every step, but take the present ambiguity as a bad omen for things to come.

Another, and trickier, problem arises from *orta*, or ‘arising from’ to translate the participle into the present. Suppose that the body is of uniform density, so the quantity of material arises from the product of the density and the volume. But it is not defined to be equal to it. Here there are two issues. First of all, is there a constant factor that has been concealed? And secondly, why this circumlocution? Cannot Newton simply say that the mass is the product of the density and the volume?

As to the second of these points, for Newton a real number is the ratio of the lengths of two real line segments. If one of these is an inch, you get the length in inches. The product of two real numbers is the ratio of the area of two rectangles, similarly with three real numbers and cuboids, and you cannot multiply four real numbers. Such was the influence of Euclid. Though Newton is happy to abandon Euclid, and consider generalised infinite power series with irrational exponents when necessary. But he has a coyness about multiplication. What do you mean by the product of the density and the volume? But he overcomes his scruples when neccessary, and at one point measures work in inch-ounces.

Finally, the term defined, namely *quantitas materiae*, is one of many used by Newton for mass. He tends to choose different names for the same concept in different contexts, like the different names used for the same person in a Russian novel. And he is loathe to coin new terms, especially in Latin. So he gets accused of not having a concept because he has no word for it.

I think that many commentators go astray by failing to appreciate the looseness of Newton’s style. His use of elegant variation means that he is highly inconsistent with his phraseology, and attempts to discern nuances of meaning in variation of language have to be assessed with great care and sensitivity. And, in the other direction, he will use the same word with different meanings, sometimes in the same sentence. He has two words that correspond, roughly, to ‘speed’, namely *velocitas* and *celeritas*. Cohen in his lengthy guide that precedes the Cohen-Whitman translation, states that they translate *velocitas* as ‘velocity’ and *celeritas* as ‘speed’, while being aware that these are not the meanings that Newton attaches to these terms. In the modern scientific parlance, ‘speed’ means speed in the sense of ‘thirty miles an hour’, and ‘velocity’ means ‘speed in a specified direction’. So speed is a scalar and velocity is a vector.

Cohen and Whitman have a strong and consistent bias towards translating Latin words by the etymologically closest English word, regardless of meaning (as Cohen admits with *velocitas* and *celeritas*). But it is not a question of vectors and scalars. Newton sometimes uses *velocitas* to mean ‘acceleration’. Moreover, he repeats an observation, but with *velocitas* at one occurrence and *celeritas* at the other. Is the translator to reproduce Newton’s elegant variation, or to convey what he meant?

One option is to choose an option at random, and to give the other in a note. But now the translation is weighed down with such notes. My own view is that, as Newton does not care which word he uses, neither should the reader.

So I intend in my next blog to turn to an example of Newton’s stunning brilliance, and leave this pedantry behind.

Charles Leedham-Green

C.R.Leedham-Green@qmul.ac.uk

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My new translation of The Principia is published by Cambridge University Press