Mathematics Books

"Have not the small Particles of Bodies certain Powers, Virtues, or Forces, by which they act at a distance, not only upon the Rays of Light for reflecting, refracting, and inflecting them, but also upon one another for producing a great Part of the Phenomena of Nature?"

"By this way of Analysis we may proceed from Compounds to Ingredients, and from Motions to the Forces producing them; and in general, from Effects to their Causes, and from particular Causes to more general ones, till the Argument end in the most general. This is the Method of Analysis: and the Synthesis consists in assuming the Causes discover'd, and establish'd as Principles, and by them explaining the Phænomena proceeding from them, and proving the Explanations."

"Now the smallest Particles of Matter may cohere by the strongest Attractions, and compose bigger Particles of weaker Virtue; and many of these may cohere and compose bigger Particles whose Virtue is still weaker, and so on for divers Successions, until the Progression end in the biggest Particles on which the Operations in Chymistry, and the Colours of natural Bodies depend, and which by cohering compose Bodies of a sensible Magnitude."

"[T]he Instinct of Brutes and Insects, can be the effect of nothing else than the Wisdom and Skill of a powerful ever-living Agent."

"How these Attractions may be perform'd, I do not here consider. What I call Attraction may be perform'd by impulse, or by some other means unknown to me. I use that Word here to signify only in general any Force by which Bodies tend towards one another, whatsoever be the Cause. For we must learn from the Phaenomena of Nature what Bodies attract one another, and what are the Laws and Properties of the Attraction, before we enquire the Cause by which the Attraction is perform'd. The Attractions of Gravity, Magnetism and Electricity, react to very sensible distances, and so have been observed by vulgar Eyes, and there may be others which reach to so small distances as hitherto escape Observation; and perhaps electrical Attraction may react to such small distances, even without being excited by Friction."

"And since Space is divisible in infinitum, and Matter is not necessarily in all places, it may be also allow'd that God is able to create Particles of Matter of several Sizes and Figures, and in several Proportions to Space, and perhaps of different Densities and Forces, and thereby to vary the Laws of Nature, and make Worlds of several sorts in several Parts of the Universe."

"As in Mathematicks, so in Natural Philosophy, the Investigation of difficult Things by the Method of Analysis, ought ever to precede the Method of Composition."

"There are therefore Agents in Nature able to make the Particles of Bodies stick together by very strong Attractions. And it is the Business of experimental Philosophy to find them out."

"[I]t seems probable to me that God, in the Beginning, formed Matter in solid, massy, hard, impenetrable, moveable Particles, of such Sizes and Figures, and with such other Properties, and in such Proportion to Space, as most conduc'd to the end for which he form'd them; and that these primitive Particles, being Solids, are incomparably harder than any porous Bodies compounded of them; even so very hard as never to wear or break in pieces; no ordinary Power being able to divide what God himself made one in the first Creation. While the Particles continue entire, they may compose Bodies of one and the same Nature and Texture in all Ages: But should they wear away, or break in pieces, the Nature of Things depending on them, would be changed."

"[W]e are not to consider the World as the Body of God, or the several Parts thereof, as the Parts of God. He is an uniform Being, void of Organs, Members or Parts, and they are his Creatures subordinate to him, and subservient to his Will."

"[A]ll material Things seem to have been composed of the hard and solid Particles above-mention'd, variously associated in the first Creation by the Counsel of an intelligent Agent. For it became him who created them to set them in order. And if he did so, it's unphilosophical to seek for any other Origin of the World, or to pretend that it might arise out of a Chaos by the mere Laws of Nature."

"[T]o derive two or three general Principles of Motion from Phænomena, and afterwards to tell us how the Properties and Actions of all corporeal Things follow from those manifest Principles, would be a very great step in Philosophy."

"Are not gross Bodies and Light convertible into one another, and may not Bodies receive much of their Activity from the Particles of Light which enter into their Composition? ... The changing of Bodies into Light, and Light into Bodies, is very conformable to the Course of Nature, which seems delighted with Transmutations."

"[D]oes it not appear from Phænomena that there is a Being incorporeal, living, intelligent, omnipresent, who in infinite Space, as it were in his Sensory, sees the things themselves intimately, and throughly perceives them, and comprehends them wholly by their immediate presence to himself."

"[T]o make way for the regular and lasting Motions of the Planets and Comets, it's necessary to empty the Heavens of all Matter, except perhaps some very thin Vapours, Steams or Effluvia, arising from the Atmospheres of the Earth, Planets and Comets, and from such an exceedingly rare Æthereal Medium...A dense Fluid can be of no use for explaining the Phænomena of Nature, the Motions of the Planets and Comets being better explain'd without it. It serves only to disturb and retard the Motions of those great Bodies, and make the frame of Nature languish: And in the Pores of Bodies, it serves only to stop the vibrating Motions of their Parts, wherein their Heat and Activity consists. And as it is of no use, and hinders the Operations of Nature, and makes her languish, so there is no evidence for its Existence, and therefore it ought to be rejected. And if it be rejected, the Hypotheses that Light consists in Pression or Motion propagated through such a Medium, are rejected with it."

"Doth not this Æthereal Medium in passing out of Water, Glass, Crystal, and other compact and dense Bodies into empty Spaces, grow denser and denser by degrees, and by that means refract the Rays of Light not in a point, but by bending them gradually in curve Lines? And doth not the gradual condensation of this Medium extend to some distance from the Bodies, and thereby cause the Inflexions of the Rays of Light, which pass by the edges of dense Bodies, at some distance from the Bodies?"

"[T]he main Business of natural Philosophy is to argue from Phænomena without feigning Hypotheses, and to deduce Causes from Effects, till we come to the very first Cause, which certainly is not mechanical; and not only to unfold the Mechanism of the World, but chiefly to resolve these and such like Questions."

"[U]nless they be exceeding rare, a great Objection arises from the regular and very lasting Motions of the Planets and Comets in all manner of Courses through the Heavens."

"Are not all Hypotheses erroneous in which Light is supposed to consist of Pression or Motion propagated through a fluid medium?"

"And for rejecting such a Medium, we have the authority of those the oldest and most celebrated philosophers of ancient Greece and Phoenicia, who made a Vacuum and Atoms and the Gravity of Atoms the first Principles of their Philosophy, tacitly attributing Gravity to some other Cause than dense Matter. Later Philosophers banish the Consideration of such a Cause out of natural Philosophy, feigning Hypotheses for explaining all things mechanically, and referring other Causes to Metaphysicks: Whereas the main Business of natural Philosophy is to argue from Phænomena without feigning Hypotheses, and to deduce Causes from Effects, till we come to the very first Cause, which certainly is not mechanical; and not only to unfold the Mechanism of the World, but chiefly to resolve these such such like Questions."

"Are not the rays of Light very small Bodies emitted from shining Substances? ... Pellucid Substances act upon Rays of Light at a distance in refracting, reflecting, and inflecting them, and the Rays mutually agitate the Parts of those Substances at a distance for heating them; and this Action and Re-action at a distance very much resembles an attractive Force between Bodies."

"How came the Bodies of Animals to be contrived with so much Art, and for what ends were their several Parts? Was the Eye contrived without Skill in Opticks, and the Ear without Knowledge of Sounds? How do the Motions of the Body follow from the Will, and whence is the Instinct in Animals? Is not the Sensory of Animals that place to which the sensitive Substance is present, and into which the sensible Species of Things are carried through the Nerves and Brain, that there they may be perceived by their immediate presence to that Substance? And these things being rightly dispatch'd, does it not appear from Phænomena that there is a Being incorporeal, living, intelligent, omnipresent, who in infinite Space, as it were in his Sensory, sees the things themselves intimately, and throughly perceives them, and comprehends them wholly by their immediate presence to himself: Of which things the Images only carried through the Organs of Sense into our little Sensoriums, are there seen and beheld by that which in us perceives and thinks. And though every true Step made in this Philosophy brings us not immediately to the Knowledge of the first Cause, yet it brings us nearer to it, and on that account is to be highly valued."

"Is not this Medium [æther] much rarer within the dense Bodies of the Sun, Stars, Planets, and Comets, than in the empty celestial Spaces between them? And in passing from them to great distances, doth it not grow denser and denser perpetually, and thereby cause the gravity of those great Bodies towards one another, and of their parts towards the Bodies; every Body endeavouring to go from the denser parts of the Medium towards the rarer? ... And though this Increase of density may at great distances be exceeding stow, yet if the elastick force of this Medium be exceeding great, it may suffice to impel Bodies from the denser parts of the Medium towards the rarer, with all that power which we call Gravity. And that the elastic force of this Medium is exceeding great, may be gather'd from the swiftness of its Vibrations."

"Do not several sorts of Rays make Vibrations of several bignesses, which according to their bigness excite Sensations of several Colours, much after the manner that the Vibrations of the Air, according to their several bignesses excite Sensations of several Sounds? And particularly do not the most refrangible Rays excite the shortest Vibrations for making a Sensation of deep violet, the least refrangible the largest form making a Sensation of deep red, and several intermediate sorts of Rays, Vibrations of several intermediate bignesses to make Sensations of several intermediate Colours?"

"As Attraction is stronger in small Magnets than in great ones in proportion to their Bulk, and Gravity is greater in the Surfaces of small Planets than in those of great ones in proportion to their bulk, and small Bodies are agitated much more by electric attraction than great ones; so the smallness of the Rays of Light may contribute very much to the power of the Agent by which they are refracted."

"Do not the Rays of Light which fall upon Bodies, and are reflected or refracted, begin to bend before they arrive at the Bodies; and are they not reflected, refracted, and inflected, by one and the same Principle, acting variously in various Circumstances?"

"To explain all nature is too difficult a task for any one man or even for any one age. 'Tis much better to do a little with certainty, & leave the rest for others that come after you, than to explain all things by conjecture without making sure of any thing."

"Do not Bodies act upon Light at a distance, and by their action bend its Rays; and is not this action (caeteris paribus) [all else being equal] strongest at the least distance?"

"Do not Bodies and Light act mutually upon one another; that is to say, Bodies upon Light in emitting, reflecting, refracting and inflecting it, and Light upon Bodies for heating them, and putting their parts into a vibrating motion wherein heat consists?"

"What is there in places empty of matter? and Whence is it that the sun and planets gravitate toward one another without dense matter between them? Whence is it that Nature doth nothing in vain? and Whence arises all that order and beauty which we see in the world? To what end are comets, and whence is it that Planets move all one and the same way in Oorbs concentrick, while comets move all manner of ways in Orbs very excentrick; and what hinders the fix'd Stars from falling upon one another?"

"And so if any one would suppose that Æther (like our Air) may contain Particles which endeavour to recede from one another (for I do not know what this Æther is) and that its Particles are exceedingly smaller than those of Air, or even than those of Light: The exceeding smallness of its Particles may contribute to the greatness of the force by which those Particles may recede from one another, and thereby make that Medium exceedingly more rare and elastick than Air, and by consequence exceedingly less able to resist the motions of projectiles, and exceedingly more able to press upon gross Bodies, by endeavouring to expand it self."

"My Design in this Book is not to explain the Properties of Light by Hypotheses, but to propose and prove them by Reason and Experiments: In order to which I shall premise the following Definitions and Axioms."

"Do not the Rays which differ in Refrangibility differ also in Flexibility; and are they not by their different inflexions separated from one another, so as after separation to make the Colours in the three Fringes...? And after what manner are they inflected to make those Fringes?"

"Are not the Rays of Light in passing by the edges and sides of Bodies, bent several times backwards and forwards, with a motion like that of an Eel? And do not the three Fringes of colour'd Light...arise from three such bendings?"

"Is not Fire a Body heated so hot as to emit Light copiously? For what else is a red hot Iron than Fire? And what else is a burning Coal than red hot Wood?"

"Do not great Bodies conserve their heat the longest, their parts heating one another, and may not great dense and fix'd Bodies, when heated beyond a certain degree, emit Light so copiously, as by the Emission and Re-action of its Light, and the Reflexions and Refractions of its Rays within its Pores to grow still hotter, till it comes to a certain period of heat, such as is that of the Sun?"

"Is not the Heat of the warm Room convey'd through the Vacuum by the Vibrations of a much subtiler Medium than Air, which after the Air was drawn out remained in the Vacuum? And is not this Medium the same with that Medium by which Light is refracted and reflected and by whose Vibrations Light communicates Heat to Bodies, and is put into Fits of easy Reflexion and easy Transmission? ... And do not hot Bodies communicate their Heat to contiguous cold ones, by the Vibrations of this Medium propagated from them into the cold ones? And is not this Medium exceedingly more rare and subtile than the Air, and exceedingly more elastick and active? And doth it not readily pervade all Bodies? And is it not (by its elastick force) expanded through all the Heavens"

"The external or relative conducibility of a solid body depends on the facility with which heat penetrates the surface, and passes from this body into a given medium, or... from the medium into the solid. The last property is modified by the... polished state of the surface... also according to the medium in which... immersed; but the interior conducibility can change only with the nature of the solid."

"The proper or interior conducibility of a body expresses the facility with which heat is propagated in passing from one internal molecule to another."

"These three elementary qualities are represented... by constant[s], and the theory... indicates experiments suitable for measuring their values. As soon as... determined... problems relating to the propagation of heat depend only on numerical analysis."

"[T]he effects of the propagation of heat depend in... every solid substance, on three elementary qualities... its capacity for heat, its own conducibility, and the exterior conducibility."

"[D]uring... cooling... heat escapes, at each instant, through the external surface, and passes into the medium... [and] this quantity is not constant; it is greatest at the beginning of... cooling. If... we consider the variable state of the internal spherical surface... [at] radius... x... there must be at each instant a... quantity of heat which traverses that surface, and passes through that part... more distant from the centre. This continuous flow of heat is variable like that through the external surface, and both are quantities comparable with each other; their ratios are numbers whose varying values are functions of the distance x, and of the time t... elapsed. It is required to determine these functions."

"[I]f two bodies of the same volume and of different nature have equal temperatures, and if the same quantity of heat be added to them, the increments of temperature are not the same; the ratio of these increments is the, ratio of their capacities for heat."

"[T]here is no mathematical theory which has a closer relation... with public economy, since it serves to give clearness and perfection to the practice of the numerous arts... founded on... heat."

"Consider next the variable state... as soon as the [heat] source has been removed; denote by t the time... passed since the... source [removal], and by v the... temperature at... m after the time t. v will be a... function F(x, t) of the distance x and the time t; the object... is to discover this function F(x, t), of which we only know as yet that the initial value... f(x) = F(x, o)."

"Let z be the constant temperature at point m [on] the mean circumference [of the ring], x the distance of this point from the [heat] source [point o], that is to say the length of the arc of the mean circumference, included between the point m and the point o... z is the highest temperature which the point m can attain by virtue of the constant action of the source, and this permanent temperature z is function f(x) of the distance x. The first part of the problem consists in determining the function f(x) which represents the permanent [temperature] state of the solid."

"If we place a solid homogeneous... sphere or cube, in a medium... [of] constant temperature... for a... long time, it will acquire at all its points... [the] temperature... of the fluid. Suppose the mass to be withdrawn... to transfer... to a cooler medium, heat will begin to be dissipated at its surface; the temperatures at different points of the mass will not be... the same, and if we suppose it divided into an infinity of layers by surfaces parallel to its external surface, each of those layers will transmit, at each instant, a certain quantity of heat to the layer which surrounds it. If... each molecule carries a separate thermometer... the state of the solid will from time to time be represented by the variable system of... these thermometric heights. It is required to express the successive states by analytical formulae, so that we may know at any... instant the temperatures... and compare the quantities of heat which flow during the same instant, between two adjacent layers, or into the surrounding medium."

"[T]he thickness of the ring is supposed... sufficiently small for the temperature to be... equal at all points of the same section perpendicular to the mean circumference."

"When the [heat] source is removed, the line which bounds the ordinates... at the different points will change its form continually."

"If now the source be suppressed, heat will continue to be propagated in the [ring's] interior... but that... lost in the... void, will no longer be compensated... by the... source, so that all... temperatures will... diminish... until... equal to the temperatures of the surrounding medium."