…continental blocks can join and rift at random.… The fact that two provinces of the Canadian Shield have been together during post-Cambrian time does not necessarily mean that they were formed close together or that the sediments lying in one province were derived from the province now beside it.

[Relativist] Rel. There is a well-known proposition of Euclid which states that “Any two sides of a triangle are together greater than the third side.” Can either of you tell me whether nowadays there is good reason to believe that this proposition is true?
[Pure Mathematician] Math. For my part, I am quite unable to say whether the proposition is true or not. I can deduce it by trustworthy reasoning from certain other propositions or axioms, which are supposed to be still more elementary. If these axioms are true, the proposition is true; if the axioms are not true, the proposition is not true universally. Whether the axioms are true or not I cannot say, and it is outside my province to consider.

Copernicus, who rightly did condemn
This eldest systeme, form’d a wiser scheme;
In which he leaves the Sun at Rest, and rolls
The Orb Terrestial on its proper Poles;
Which makes the Night and Day by this Career,
And by its slow and crooked Course the Year.
The famous Dane, who oft the Modern guides,
To Earth and Sun their Provinces divides:
The Earth’s Rotation makes the Night and Day,
The Sun revolving through th’ Eccliptic Way
Effects the various seasons of the Year,
Which in their Turn for happy Ends appear.
This Scheme or that, which pleases best, embrace,
Still we the Fountain of their Motion trace.
Kepler asserts these Wonders may be done
By the Magnetic Vertue of the Sun,
Which he, to gain his End, thinks fit to place
Full in the Center of that mighty Space,
Which does the Spheres, where Planets roll, include,
And leaves him with Attractive Force endu’d.
The Sun, thus seated, by Mechanic Laws,
The Earth, and every distant Planet draws;
By which Attraction all the Planets found
Within his reach, are turn'd in Ether round.

A century ago astronomers, geologists, chemists, physicists, each had an island of his own, separate and distinct from that of every other student of Nature; the whole field of research was then an archipelago of unconnected units. To-day all the provinces of study have risen together to form a continent without either a ferry or a bridge.

All that comes above that surface [of the globe] lies within the province of Geography. All that comes below that surface lies inside the realm of Geology. The surface of the earth is that which, so to speak, divides them and at the same time “binds them together in indissoluble union.” We may, perhaps, put the case metaphorically. The relationships of the two are rather like that of man and wife. Geography, like a prudent woman, has followed the sage advice of Shakespeare and taken unto her “an elder than herself;” but she does not trespass on the domain of her consort, nor could she possibly maintain the respect of her children were she to flaunt before the world the assertion that she is “a woman with a past.”

But by far the greatest obstacle to the progress of science and to the undertaking of new tasks and provinces therein is found in this—that men despair and think things impossible.

But when science, passing beyond its own limits, assumes to take the place of theology, and sets up its own conception of the order of Nature as a sufficient account of its cause, it is invading a province of thought to which it has no claim, and not unreasonably provokes the hostility of its best friends.

Definition of Mathematics.—It has now become apparent that the traditional field of mathematics in the province of discrete and continuous number can only be separated from the general abstract theory of classes and relations by a wavering and indeterminate line. Of course a discussion as to the mere application of a word easily degenerates into the most fruitless logomachy. It is open to any one to use any word in any sense. But on the assumption that “mathematics” is to denote a science well marked out by its subject matter and its methods from other topics of thought, and that at least it is to include all topics habitually assigned to it, there is now no option but to employ “mathematics” in the general sense of the “science concerned with the logical deduction of consequences from the general premisses of all reasoning.”

I confess that I have as vast contemplative ends, as I have moderate civil ends: for I have taken all knowledge to be my province.

I have learnt that all our theories are not Truth itself, but resting places or stages on the way to the conquest of Truth, and that we must be contented to have obtained for the strivers after Truth such a resting place which, if it is on a mountain, permits us to view the provinces already won and those still to be conquered.

I have repeatedly had cause to refer to certain resemblances between the phenomena of irritability in the vegetable kingdom and those of the animal body, thus touching a province of investigation which has hitherto been far too little cultivated. In the last instance, indeed, I might say animal and vegetable life must of necessity agree in all essential points, including the phenomena of irritability also, since it is established that the animal organism is constructed entirely and simply from the properties of these substances that all vital movements both of plants and animals are to be explained.

I now never make the preparations for penetrating into some small province of nature hitherto undiscovered without breathing a prayer to the Being who hides His secrets from me only to allure me graciously on to the unfolding of them.

I observed on most collected stones the imprints of innumerable plant fragments which were so different from those which are growing in the Lyonnais, in the nearby provinces, and even in the rest of France, that I felt like collecting plants in a new world… The number of these leaves, the way they separated easily, and the great variety of plants whose imprints I saw, appeared to me just as many volumes of botany representing in the same quarry the oldest library of the world.

In the animal world, on the other hand, the process of evolution is characterised by the progressive discrimination of the animal and vegetative functions, and a consequent differentiation of these two great provinces into their separate departments.

It has been said that no science is established on a firm basis unless its generalisations can be expressed in terms of number, and it is the special province of mathematics to assist the investigator in finding numerical relations between phenomena. After experiment, then mathematics. While a science is in the experimental or observational stage, there is little scope for discerning numerical relations. It is only after the different workers have “collected data” that the mathematician is able to deduce the required generalisation. Thus a Maxwell followed Faraday and a Newton completed Kepler.

It is the province of knowledge to speak and it is the privilege of wisdom to listen.

Nature is a vast tablet, inscribed with signs, each of which has its own significancy, and becomes poetry in the mind when read; and geology is simply the key by which myriads of these signs, hitherto indecipherable, can be unlocked and perused, and thus a new province added to the poetical domain.

Piecemeal social engineering resembles physical engineering in regarding the ends as beyond the province of technology. (All that technology may say about ends is whether or not they are compatible with each other or realizable.)

Poincaré was the last man to take practically all mathematics, pure and applied, as his province. … Few mathematicians have had the breadth of philosophic vision that Poincaré had, and none is his superior in the gift of clear exposition.

Religion has been compelled by science to give up one after another of its dogmas—of those assumed cognitions which it could not substantiate. In the mean time, Science substituted for the personalities to which Religion ascribed phenomena certain metaphysical entities; and in doing this it trespassed on the province of religion; since it classed among the things which it comprehended certain forms of the incomprehensible.

Science can point out dangers, but science cannot turn the direction of minds and hearts. That is the province of spiritual powers within and without our very beings—powers that are the mysteries of life itself.

Science gives us the grounds of premises from which religious truths are to be inferred; but it does not set about inferring them, much less does it reach the inference; that is not its province. It brings before us phenomena, and it leaves us, if we will, to call them works of design, wisdom, or benevolence; and further still, if we will, to proceed to confess an Intelligent Creator. We have to take its facts, and to give them a meaning, and to draw our own conclusions from them. First comes Knowledge, then a view, then reasoning, then belief. This is why Science has so little of a religious tendency; deductions have no power of persuasion. The heart is commonly reached, not through the reason, but through the imagination, by means of direct impressions, by the testimony of facts and events, by history, by description. Persons influence us, voices melt us, looks subdue us, deeds inflame us. Many a man will live and die upon a dogma; no man will be a martyr for a conclusion.

Science is the flower of the altruism of the ages, by which nothing that lives “liveth for itself alone.” The recognition of facts and laws is the province of science.

The Analytical Engine has no pretensions whatever to originate anything. It can do whatever we know how to order it to perform. It can follow analysis; but it has no power of anticipating any analytical relations or truths. Its province is to assist us to making available what we are already acquainted with.
[Describing Charles Babbage's machine.]

The antagonism between science and religion, about which we hear so much, appears to me purely factitious, fabricated on the one hand by short-sighted religious people, who confound theology with religion; and on the other by equally short-sighted scientific people who forget that science takes for its province only that which is susceptible of clear intellectual comprehension.

The complacent manner in which geologists have produced their theories has been extremely amusing; for often with knowledge (and that frequently inaccurate) not extending beyond a given province, they have described the formation of a world with all the detail and air of eye-witnesses. That much good ensues, and that the science is greatly advanced, by the collision of various theories, cannot be doubted. Each party is anxious to support opinions by facts. Thus, new countries are explored, and old districts re-examined; facts come to light that do not suit either party; new theories spring up; and, in the end, a greater insight into the real structure of the earth's surface is obtained.

The determining cause of most wars in the past has been, and probably will be of all wars in the future, the uncertainty of the result; war is acknowledged to be a challenge to the Unknown, it is often spoken of as an appeal to the God of Battles. The province of science is to foretell; this is true of every department of science. And the time must come—how soon we do not know—when the real science of war, something quite different from the application of science to the means of war, will make it possible to foresee with certainty the issue of a projected war. That will mark the end of battles; for however strong the spirit of contention, no nation will spend its money in a fight in which it knows it must lose.

The errors of a wise man are literally more instructive than the truths of a fool. The wise man travels in lofty, far-seeing regions; the fool in low-lying, high-fenced lanes; retracing the footsteps of the former, to discover where he diviated, whole provinces of the universe are laid open to us; in the path of the latter, granting even that he has not deviated at all, little is laid open to us but two wheel-ruts and two hedges.

The field of scientific abstraction encompasses independent kingdoms of ideas and of experiments and within these, rulers whose fame outlasts the centuries. But they are not the only kings in science. He also is a king who guides the spirit of his contemporaries by knowledge and creative work, by teaching and research in the field of applied science, and who conquers for science provinces which have only been raided by craftsmen.

The ideal of mathematics should be to erect a calculus to facilitate reasoning in connection with every province of thought, or of external experience, in which the succession of thoughts, or of events can be definitely ascertained and precisely stated. So that all serious thought which is not philosophy, or inductive reasoning, or imaginative literature, shall be mathematics developed by means of a calculus.

The laboratory work was the province of Dr Searle, an explosive, bearded Nemesis who struck terror into my heart. If one made a blunder one was sent to ‘stand in the corner’ like a naughty child. He had no patience with the women students. He said they disturbed the magnetic equipment, and more than once I heard him shout ‘Go and take off your corsets!’ for most girls wore these garments then, and steel was beginning to replace whalebone as a stiffening agent. For all his eccentricities, he gave us excellent training in all types of precise measurement and in the correct handling of data.

The position of the anthropologist of to-day resembles in some sort the position of classical scholars at the revival of learning. To these men the rediscovery of ancient literature came like a revelation, disclosing to their wondering eyes a splendid vision of the antique world, such as the cloistered of the Middle Ages never dreamed of under the gloomy shadow of the minster and within the sound of its solemn bells. To us moderns a still wider vista is vouchsafed, a greater panorama is unrolled by the study which aims at bringing home to us the faith and the practice, the hopes and the ideals, not of two highly gifted races only, but of all mankind, and thus at enabling us to follow the long march, the slow and toilsome ascent, of humanity from savagery to civilization. And as the scholar of the Renaissance found not merely fresh food for thought but a new field of labour in the dusty and faded manuscripts of Greece and Rome, so in the mass of materials that is steadily pouring in from many sides—from buried cities of remotest antiquity as well as from the rudest savages of the desert and the jungle—we of to-day must recognise a new province of knowledge which will task the energies of generations of students to master.

The ravages committed by man subvert the relations and destroy the balance which nature had established between her organized and her inorganic creations; and she avenges herself upon the intruder, by letting loose upon her defaced provinces destructive energies hitherto kept in check by organic forces destined to be his best auxiliaries, but which he has unwisely dispersed and driven from the field of action. When the forest is gone, the great reservoir of moisture stored up in its vegetable mould is evaporated, and returns only in deluges of rain to wash away the parched dust into which that mould has been converted. The well-wooded and humid hills are turned to ridges of dry rock, which encumbers the low grounds and chokes the watercourses with its debris, and–except in countries favored with an equable distribution of rain through the seasons, and a moderate and regular inclination of surface–the whole earth, unless rescued by human art from the physical degradation to which it tends, becomes an assemblage of bald mountains, of barren, turfless hills, and of swampy and malarious plains. There are parts of Asia Minor, of Northern Africa, of Greece, and even of Alpine Europe, where the operation of causes set in action by man has brought the face of the earth to a desolation almost as complete as that of the moon; and though, within that brief space of time which we call “the historical period,” they are known to have been covered with luxuriant woods, verdant pastures, and fertile meadows, they are now too far deteriorated to be reclaimable by man, nor can they become again fitted for human use, except through great geological changes, or other mysterious influences or agencies of which we have no present knowledge, and over which we have no prospective control. The earth is fast becoming an unfit home for its noblest inhabitant, and another era of equal human crime and human improvidence, and of like duration with that through which traces of that crime and that improvidence extend, would reduce it to such a condition of impoverished productiveness, of shattered surface, of climatic excess, as to threaten the depravation, barbarism, and perhaps even extinction of the species.

There are problems to whose solution I would attach an infinitely greater importance than to those of mathematics, for example touching ethics, or our relation to God, or concerning our destiny and our future; but their solution lies wholly beyond us and completely outside the province of science.

There are three stages in the development of science: First, there is the observation of things and facts—the scientists map out and inventory the objects in each department of Nature; secondly, the interrelations are investigated, and this leads to a knowledge of forces and influences which produce or modify those objects…. This is the dynamic stage, the discovery of forces and laws connecting each fact with all other facts, and each province of Nature with all other provinces of Nature. The goal of this second stage of science is to make each fact in Nature throw light on all the other facts, and thus to illuminate each by all. … Science in its third and final stage learns to know everything in Nature as a part of a process which it studies in the history of its development. When it comes to see each thing in the perspective of its evolution, it knows it and comprehends it.

There is only one law of Nature—the second law of thermodynamics—which recognises a distinction between past and future more profound than the difference of plus and minus. It stands aloof from all the rest. … It opens up a new province of knowledge, namely, the study of organisation; and it is in connection with organisation that a direction of time-flow and a distinction between doing and undoing appears for the first time.

We stand by the river and admire the great body of water flowing so sweetly on; could you trace it back to its source, you might find a mere rivulet, but meandering on, joined by other streams and by secret springs, and fed by the rains and dews of heaven, it gathers volume and force, makes its way through the gorges of the mountains, plows, widens and deepens its channel through the provinces, and attains its present majesty.