Leibniz to Newton

Hanover, 7 March 1692/3

How great I think the debt owed to you, by our knowledge of mathematics and of all nature, I have acknowledged in public also when occasion offered. You had given an astonishing development to geometry by your series; but when you published your work, the Principia, you showed that even what is not subject to the received analysis is an open book to you. I too have tried by the application of convenient symbols, which exhibit differences and sums, to submit that geometry which I call ‘transcendent’ in some sense to analysis, and the attempt did not go badly. But to put the last touches I am still looking for something big from you, first how best problems which seek lines from a given property of their tangents, may be reduced to squarings, and next how the squarings themselves – and this is what I would like very much to see – may be reduced to the rectifications of curves, simpler in all cases than the measurings of surfaces or volumes.

But above all I would wish that, perfected in geometrical problems, you would continue, as you have begun, to handle nature in mathematical terms; and in this field you have by yourself with very few companions gained an immense return for your labour. You have made the astonishing discovery that Kepler’s ellipses result simply from the conception of attraction or gravitation and passage in a planet. And yet I would incline to believe that all these are caused or regulated by the motion of a fluid medium, on the analogy of gravity and magnetism as we know it here. Yet this solution would not at all detract from the value and truth of your discovery. …

Since the ancients (according to Pappus) considered mechanics to be of the greatest importance in the investigation of nature and science and since the moderns – rejecting substantial forms and occult qualities – have undertaken to reduce the phenomena of nature to mathematical laws, it has seemed best in this treatise to concentrate on mathematics as it relates to natural philosophy. The ancients divided mechanics into two parts: the rational, which proceeds rigorously through demonstrations, and the practical. Practical mechanics is the subject that comprises all the manual arts, from which the subject of mechanics as a whole has adopted its name. But since those who practise an art do not generally work with a high degree of exactness, the whole subject of mechanics is distinguished from geometry by the attribution of exactness to geometry and of anything less than exactness to mechanics. Yet the errors do not come from the art but from those who practise the art. Anyone who works with less exactness is a more imperfect mechanic, and if anyone could work with the greatest exactness, he would be the most perfect mechanic of all. For the description of straight lines and circles, which is the foundation of geometry, appertains to mechanics. Geometry does not teach how to describe these straight lines and circles, but postulates such a description. For geometry postulates that a beginner has learned to describe lines and circles exactly before he approaches the threshold of geometry, and then it teaches how problems are solved by these operations.

Newton as natural philosopher

Isaac Newton’s influence is ubiquitous 300 years after his death. We employ Newtonian mechanics in a wide range of cases, students worldwide learn the calculus that he co-discovered with Leibniz, and the law of universal gravitation characterizes what is still considered a fundamental force. Indeed, the idea that a force can be “fundamental,” irreducible to any other force or phenomenon in nature, is largely due to Newton, and still has currency in the twenty-first century. Remarkably, Newton’s status as a theorist of motion and of forces, and his work as a mathematician, is equaled by his status as an unparalleled experimentalist.

The philosophy which Mr Newton in his Principles and Opticks has pursued is experimental; and it is not the business of experimental philosophy to teach the causes of things any further than they can be proved by experiments. We are not to fill this philosophy with opinions which cannot be proved by phenomena. In this philosophy hypotheses have no place, unless as conjectures or questions proposed to be examined by experiments. For this reason, Mr Newton in his Opticks distinguished those things which were made certain by experiments from those things which remained uncertain, and which he therefore proposed in the end of his Opticks in the form of queries. For this reason, in the preface to his Principles, when he had mentioned the motions of the planets, comets, moon and sea as deduced in this book from gravity, he added: ‘If only we could derive the other phenomena of nature from mechanical principles by the same kind of reasoning! For many things lead me to have a suspicion that all phenomena may depend on certain forces by which the particles of bodies, by causes not yet known, either are impelled towards one another and cohere in regular figures, or are repelled from one another and recede. Since these forces are unknown, philosophers have hitherto made trial of nature in vain.’ And in the end of this book in the second edition, he said that for want of a sufficient number of experiments, he forbore to describe the laws of the actions of the spirit or agent by which this attraction is performed. And for the same reason he is silent about the cause of gravity, there occurring no experiments or phenomena by which he might prove what was the cause thereof. And this he hath abundantly declared in his Principles, near the beginning thereof, in these words: ‘I am not now considering the physical causes and sites of forces’ [definition 8].

Cotes to Bentley

10 March 1713

Sir,

I received what you wrote to me in Sir Isaac’s letter. I will set about the index in a day or two. As to the preface I should be glad to know from Sir Isaac with what view he thinks proper to have it written. You know the Book has been received abroad with some disadvantage, and the cause of it may easily be guessed at. The Commercium Epistolicum lately published by order of the Royal Society gives such indubitable proof of Mr Leibniz’s want of candour that I shall not scruple in the least to speak out the full truth of the matter if it be thought convenient. There are some pieces of his looking this way, which deserve a censure, as his Tentamen de Motuum Coelestium causis. If Sir Isaac is willing that something of this nature may be done, I should be very glad if, whilst I am making the index, he would be pleased to consider of it and put down a few notes of what he thinks most material to be insisted on. This I say upon supposition that I write the preface myself. But I think it will be much more advisable that you or he or both of you should write it whilst you are in town. You may depend upon it that I will own it, and defend it as well as I can, if hereafter there be occasion.

I am, Sir,

Your most obliged and humble Servant

ROGER COTES

19 February 1672

Sir,

To perform my late promise to you, I shall without further ceremony acquaint you, that in the beginning of the year 1666 (at which time I applied myself to the grinding of optic glasses of other figures than spherical) I procured me a triangular glass prism, to try therewith the celebrated phenomena of colours. And in order thereto having darkened my chamber, and made a small hole in my window shuts, to let in a convenient quantity of the sun’s light, I placed my prism at its entrance, that it might be thereby refracted to the opposite wall. It was at first a very pleasing divertisement, to view the vivid and intense colours produced thereby; but after a while applying myself to consider them more circumspectly, I became surprised to see them in an oblong form; which, according to the received laws of refraction, I expected should have been circular.

They were terminated at the sides with straight lines, but at the ends, the decay of light was so gradual, that it was difficult to determine justly, what was their figure; yet they seemed semicircular.

Comparing the length of this coloured spectrum with its breadth, I found it about five times greater; a disproportion so extravagant that it excited me to a more than ordinary curiosity of examining, from whence it might proceed. I could scarce think, that the various thickness of the glass, or the termination with shadow or darkness, could have any influence on light to produce such an effect; yet I thought it not amiss to examine first these circumstances, and so tried, what would happen by transmitting light through parts of the glass of diverse thicknesses, or through holes in the window of diverse bignesses, or by setting the prism without so that the light might pass through it, and be refracted before it was terminated by the hole: but I found none of these circumstances material. The fashion of the colours was in all these cases the same. …

Paper of directions given by Newton to Bentley respecting the books to be read before endeavoring to read and understand the Principia

c. July 1691

Next after Euclid’s Elements the elements of the Conic sections are to be understood. And for this end you may read either the first part of the Elementa Curvarum of John De Witt, or De la Hire’s late treatise of the conic sections, or Dr Barrow’s epitome of Apollonius.

For algebra read first Barthin’s introduction and then peruse such problems as you will find scattered up & down in the commentaries on Descartes’s Geometry and other algebraical writings of Francis Schooten. I do not mean that you should read over all those commentaries, but only the solutions of such problems as you will here & there meet with. You may meet with De Witt’s Elementa curvarum & Bartholin’s introduction bound up together with Descartes’s Geometry and Schooten’s commentaries.

For astronomy read first the short account of the Copernican system in the end of Gassendi’s Astronomy & then so much of Mercator’s Astronomy as concerns the same system & the new discoveries made in the heavens by telescopes in the appendix.

These are sufficient for understanding my book: but if you can procure Huygens’s Horologium oscillatorium, the perusal of that will make you much more ready. …

Cambridge, 28 February 1678/9

Honoured Sir,

I have so long deferred to send you my thoughts about the physical qualities we spoke of, that did I not esteem myself obliged by promise, I think I should be ashamed to send them at all. The truth is, my notions about things of this kind are so indigested, that I am not well satisfied myself in them; and what I am not satisfied in, I can scarce esteem fit to be communicated to others; especially in natural philosophy, where there is no end of fancying. But because I am indebted to you, and yesterday met with a friend, Mr Maulyverer, who told me he was going to London, and intended to give you the trouble of a visit, I could not forbear to take the opportunity of conveying this to you by him.

1. It being only an explication of qualities, which you desire of me, I shall set down my apprehensions in the form of suppositions, as follows. And first, I suppose, that there is diffused through all places an aethereal substance, capable of contraction and dilatation [i.e. dilation], strongly elastic, and in a word much like air in all respects, but far more subtle.

2. I suppose this aether pervades all gross bodies, but yet so as to land rarer in their pores than in free spaces, and so much the rarer, as their pores are less. And this I suppose (with others) to be the cause, why light incident on those bodies is refracted towards the perpendicular; why two well polished metals cohere in a receiver exhausted of air; why mercury stands sometimes up to the top of a glass pipe, though much higher than 30 inches; and one of the main causes, why the parts of all bodies cohere; also the cause of filtration, and of the rising of water in small glass pipes above the surface of the stagnating water they are dipped into: for I suspect the other may stand rarer, not only in the insensible pores of bodies, but even in the very sensible cavities of those pipes. And the same principle may cause menstruums [i.e. solvents] to pervade with violence the pores of the bodies they dissolve, that surrounding [the] aether, as well as the atmosphere, pressing them together. …

Scripture abused to prove the immoveableness of the globe of the earth

In determining the true system of the world the main question is whether the earth do rest or be moved. For deciding this some bring texts of scripture, but in my opinion misinterpreted, the scriptures speaking not in the language of astronomers (as they think) but in that of the common people to whom they were written. So where ’tis said that God hath made the round world so fast that it cannot be moved, the prophet intended not to teach mathematicians the spherical figure and immoveableness of the whole earth and sea in the heavens but to tell the vulgar in their own dialect that God had made the great continent of Asia, Europe and Africa so fast upon its foundations in the great ocean that it cannot be moved therein after the manner of a floating island. For this continent was the whole habitable world anciently known and by the ancient eastern nations was accounted round or circular, as was also the sea encompassing it. And this earth and sea they accounted flat as if the sun, moon and stars ascended out of the ocean at their rising and went down into it again at their setting. This continent is the world or earth usually mentioned in scripture and there described to be broad and to have end or borders, that is circular ones, whose centre some placed in Egypt others at Delphos, others at Jerusalem. And this world the prophets consider as established in the ocean upon sure and immoveable foundations at the first creation. The heavens were of old and the earth standing out of the water and in the water (that is in the midst of the ocean like an island) by the word of God (2 Pet. 3.5.). Thou Lord in the beginning hast laid the foundations of the earth and the heavens are the work of thine hands (Psalms 102.25, Proverbs 8.29). Where wast thou when I laid the foundations of the earth. Declare if thou hast understanding who hath laid the measures thereof or who hath stretched the line over it. Whereupon are the foundations thereof fixed or who hath laid the corner stone thereof, when the stars of the morning praised me together, etc. (Job 38.4).

It is fitting to treat the science of the weight and of the equilibrium of fluids and solids in fluids by a twofold method. To the extent that it appertains to the mathematical sciences, it is reasonable that I largely abstract it from physical considerations. And for this reason I have undertaken to demonstrate its individual propositions from abstract principles, sufficiently well known to the student, strictly and geometrically. Since this doctrine may be judged to be somewhat akin to natural philosophy, in so far as it may be applied to making clear many of the phenomena of natural philosophy and in order, moreover, that its usefulness may be particularly apparent and the certainty of its principles perhaps confirmed, I shall not be reluctant to illustrate the propositions abundantly from experiments as well, in such a way, however, that this freer method of discussion, disposed in scholia, may not be confused with the former, which is treated in lemmas, propositions and corollaries.

The foundations from which this science may be demonstrated are either definitions of certain words, or axioms and postulates no one denies. And of these I treat directly.

Definitions

The terms ‘quantity’, ‘duration’, and ‘space’ are too well known to be susceptible of definition by other words.

1. Definition 1. Place is a part of space which something fills completely.

2. Definition 2. Body is that which fills place.

3. Definition 3. Rest is remaining in the same place.