great circles, the visible angle comprehended under the former must be equal to the visible angle comprehended under the latter. But the visible angle comprehended under the two great circles, when seen from the centre, is of the same magnitude with the spherical angle which they really comprehend, as mathematicians know; there fore, the visible angle made by any two visible lines is equal to the spherical angle made by the two great circles of the sphere which are their representatives. 5. Hence it is evident, that every visible right-lined triangle will coincide in all its parts with some spherical triangle. The sides of the one will appear equal to the sides of the other, and the angles of the one to the angles of the other, each to each; and, therefore, the whole of the one triangle will appear equal to the whole of the other. In a word, to the eye they will be one and the same, and have the same mathematical properties. The properties, therefore, of visible right-lined triangles are not the same with the properties of plain triangles, but are the same with those of spherical triangles. 6. Every lesser circle of the sphere will appear a circle to the eye, placed, as we have supposed all along, in the centre of the sphere; and, on the other hand, every visible circle will appear to coincide with some lesser circle of the sphere. 7. Moreover, the whole surface of the sphere will represent the whole of visible space; for, since every visible point coincides with some point of the surface of the sphere, and has the same visible place, it follows, that all the parts of the spherical surface taken together, will represent all possible visible places-that is, the whole of visible space. And from this it follows, in the last place 8. That every visible figure will be represented by that part of the surface of the sphere on which it might be projected, the eye being in the centre. And every such visible figure will bear the same ratio to the whole of visible space, as the part of the spherical surface which represents it, bears to the whole spherical surface. The mathematical reader, I hope, will enter into these principles with perfect facility, and will as easily perceive that the following propositions witli regard to visible figure and space, which we offer only as a specimen, may be mathematically demonstrated from them, and are not less true nor less evident than the propositions of Euclid, with regard to tangible figures. Pop. 1. Every right line being produced, will at last return into itself. 2. A right line, returning into itself, is the longest possible right line; and all other right lines bear a finite ratio to it. 3. A right line returning into itself, divides the whole of visible space into two equal parts, which will both be comprehended under this right line. 4. The whole of visible space beais a finite ratio to any part of it. 5. Any two right lines being produced, will meet in two points, and mutually bisect each other 6. If two lines be parallel-that is, every where equally distant from each otherthey cannot both be straight. 7. Any right line being given, a point may be found, which is at the same distance from all the points of the given right line. 8. A circle may be parallel to a right line-that is, may be equally distant from it in all its parts. 9. Right-lined triangles that are similar, are also equal. 10. Of every right-lined triangle, the three ang es taken together, are greater than two right angles. 11. The angles of a right-lined triangle, may all be right angles, or all obtuse angles. 12. Unequal circles are not as the squares of their diameters, nor are their circumferences in the ratio of their diameters. This small specimen of the geometry of visibles, is intended to lead the reader to a clear and distinct conception of the figure and extension which is presented to the mind by vision; and to demonstrate the truth of what we have affirmed abovenamely, that those figures and that extension which are the immediate objects of sight, are not the figures and the extension about which common geometry is employed; that the geometrician, while he looks at his diagram, and demonstrates a proposition, hath a figure presented to his eye, which is only a sign and representative of a tangible figure; that he gives not the least attention to the first, but attends only to the last; and that these two figures have different properties, so that what he demonstrates of the one, is not true of the other. It deserves, however, to be remarked, that, as a small part of a spherical surface differs not sensibly from a plain surface, so a small part of visible extension differs very little from that extension in length and breadth, which is the object of touch. And it is likewise to be observed, that the human eye is so formed, that an object which is seen distinctly and at one view, can occupy but a small part of visible space; for we never see distinctly what is at a considerable distance from the axis of the eye; and, therefore, when we would see a large object at one view, the eye must be at so great a distance, that the object occupies but a small part of visible space. From these two observations, it follows, that plain figures which are seen at one view, when their planes are not oblique, but direct to the eye, differ little from the visible figures which they present to the eye. The several lines in the tangible figure, have very nearly the same proportion to each other as in the visible; and the angles of the one are very nearly, al- | though not strictly and mathematically, equal to those of the other. Although, therefore, we have found many instances of natural signs which have no similitude to the things signified, this is not the case with regard to visible figure. It hath, in all cases, such a similitude to the thing signified by it, as a plan or profile hath to that which it represents; and, in some cases, the sign and thing signified have to all sense the same figure and the same proportions. If we could find a being endued with sight only, without any other external sense, and capable of reflecting and reasoning upon what he sees, the notions and philosophical speculations of such a being, might assist us in the difficult task of distinguishing the perceptions which we have purely by sight, from those which derive their origin from other senses. Let us suppose such a being, and conceive, as well as we can, what notion he would have of visible objects, and what conclusions he would deduce from them. must not conceive him disposed by his constitution, as we are, to consider the visible appearance as a sign of something else: it is no sign to him, because there is nothing signified by it; and, therefore, we must suppose him as much disposed to attend to the visible figure and extension of bodies, as we are disposed to attend to their tangible figure and extension. We If various figures were presented to his sense, he might, without doubt, as they grow familiar, compare them together, and perceive wherein they agree, and wherein they differ. He might perceive visible objects to have length and breadth, but could have no notion of a third dimension, any more than we can have of a fourth.* All visible objects would appear to be terminated by lines, straight or curve; and objects terminated by the same visible lines, would occupy the same place, and fill the same part of visible space. It would not be possible for him to conceive one object to be behind another, or one to be nearer, another more distant. To us, who conceive three dimensions, a line may be conceived straight; or it may be conceived incurvated in one dimension, This proceeds upon the supposition that our notion of space is merely empirical.-H. | and straight in another; or, lastly, it may be incurvated in two dimensions. Suppose a line to be drawn upwards and downwards, its length makes one dimension, which we shall call upwards and downwards; and there are two dimensions remaining, according to which it may be straight or curve. It may be bent to the right or to the left; and, if it has no bending either to right or left, it is straight in this dimension. But supposing it straight in this dimension of right and left, there is still another dimension remaining, in which it may be curve; for it may be bent backwards or forwards. When we conceive a tangible straight line, we exclude curvature in either of these two dimensions: and as what is conceived to be excluded, must be conceived, as well as what is conceived to be included, it follows that all the three dimensions enter into our conception of a straight line. Its length is one dimension, its straightness in two other dimensions is included, or curvature in these two dimensions excluded, in the conception of it. The being we have supposed, having no conception of more than two dimensions, of which the length of a line is one, cannot possibly conceive it either straight or curve in more than one dimension; so that, in his conception of a right line, curvature to the right hand or left is excluded; but curvature backwards or forwards cannot be excluded, because he neither hath, nor can have any conception of such curvature. Hence we see the reason that a line, which is straight to the eye, may return into itself; for its being straight to the eye, implies only straightness in one dimension; and a line which is straight in one dimension may, notwithstanding, be curve in another dimension, and so may return into itself. To us, who conceive three dimensions, a surface is that which hath length and breadth, excluding thickness; and a surface may be either plain in this third dimension, or it may be incurvated: so that the notion of a third dimension enters into our conception of a surface; for it is only by means of this third dimension that we can distinguish surfaces into plain and curve surfaces; and neither one nor the other can be conceived without conceiving a third dimension. The being we have supposed, having no conception of a third dimension, his visible figures have length and breadth indeed; but thickness is neither included nor excluded, being a thing of which he has no conception. And, therefore, visible figures, although they have length and breadth, as surfaces have, yet they are neither plain surfaces nor curve surfaces. For a curve surface implies curvature in a third dimension, and a plain surface implies the want of curvature in a third dimension; and such a being can conceive neither of these, because he has no conception of a third dimension. Moreover, although he hath a distinct conception of the inclination of two lines which make an angle, yet he can neither conceive a plain angle nor a spherical angle. Even his notion of a point is somewhat less determined than ours. In the notion of a point, we exclude length, breadth, and thickness; he excludes length and breadth, but cannot either exclude or include thickness, because he hath no conception of it. Having thus settled the notions which such a being as we have supposed might form of mathematical points, lines, angles, and figures, it is easy to see, that, by comparing these together, and reasoning about them, he might discover their relations, and form geometrical conclusions built upon self-evident principles. He might likewise, without doubt, have the same notions of numbers as we have, and form a system of arithmetic. It is not material to say in what order he might proceed in such discoveries, or how much time and pains he might employ about them, but what such a being, by reason and ingenuity, without any materials of sensation but those of sight only, might discover. As it is more difficult to attend to a detail of possibilities than of facts, even of slender authority, I shall beg leave to give an extract from the travels of Johannes Rudolphus Anepigraphus, a Rosicrucian philosopher, who having, by deep study of the occult sciences, acquired the art of transporting himself to various sublunary regions, and of conversing with various orders of intelligences, in the course of his adventures became acquainted with an order of beings exactly such as I have supposed. How they communicate their sentiments to one another, and by what means he became acquainted with their language, and was initiated into their philosophy, as well as of many other particulars, which might have gratified the curiosity of his readers, and, perhaps, added credibility to his relation, he hath not thought fit to inform us ; these being matters proper for adepts only to know. His account of their philosophy is as follows: "The Idomenians," saith he, "are many of them very ingenious, and much given to contemplation. In arithmetic, geometry, metaphysics, and physics, they have most elaborate systems. In the two latter, indeed, they have had many disputes carried on with great subtilty, and are divided into various sects; yet in the two former there hath been no less unanimity than among the human species. Their princi ples relating to numbers and arithmetic, making allowance for their notation, differ in nothing from ours-but their geometry differs very considerably." As our author's account of the geometry of the Idomenians agrees in everything with the geometry of visibles, of which we have already given a specimen, we shall pass over it. He goes on thus :-" Colour, extension, and figure, are conceived to be the essential properties of body. A very considerable sect maintains, that colour is the essence of body. If there had been no colour, say they, there had been no perception or sensation. Colour is all that we perceive, or can conceive, that is peculiar to body; extension and figure being modes common to body and to empty space. And if we should suppose a body to be annihilated, colour is the only thing in it that can be annihilated; for its place, and consequently the figure and extension of that place, must remain, and cannot be imagined not to exist. These philosophers hold space to be the place of all bodies, immoveable and indestructible, without figure, and similar in all its parts, incapable of increase or diminution, yet not unmeasurable; for every the least part of space bears a finite ratio to the whole. So that with them the whole extent of space is the common and natural measure of everything that hath length and breadth; and the magnitude of every body and of every figure is expressed by its being such a part of the universe. In like manner, the common and natural measure of length is an infinite right line, which, as hath been before observed, returns into itself, and hath no limits, but bears a finite ratio to every other line. "As to their natural philosophy, it is now acknowledged by the wisest of them to have been for many ages in a very low state. The philosophers observing, that body can differ from another only in colour, figure, or magnitude, it was taken for granted, that all their particular qualities must arise from the various combinations of these their essential attributes; and, therefore, it was looked upon as the end of natural philosophy, to shew how the various combinations of these three qualities in different bodies produced all the phænomena of nature. It were endless to enumerate the various systems that were invented with this view, and the disputes that were carried on for ages; the followers of every system exposing the weak sides of other systems, and palliating those of their own, with great art. "At last, some free and facetious spirits, wearied with eternal disputation, and the labour of patching and propping weak systems, began to complain of the subtilty of nature; of the infinite changes that bodies undergo in figure, colour, and magnitude; | and of the difficulty of accounting for these appearances-making this a pretence for giving up all inquiries into the causes of things, as vain and fruitless. "These wits had ample matter of mirth and ridicule in the systems of philosophers; and, finding it an easier task to pull down than to build or support, and that every sect furnished them with arms and auxiliaries to destroy another, they began to spread mightily, and went on with great success. Thus philosophy gave way to seepticism and irony, and those systems which had been the work of ages, and the admiration of the learned, became the jest of the vulgar: for even the vulgar readily took part in the triumph over a kind of learning which they had long suspected, because it produced nothing but wrangling and altercation. The wits, having now acquired great reputation, and being flushed with success, began to think their triumph incomplete, until every pretence to knowledge was overturned; and accordingly began their attacks upon arithmetic, geometry, and even upon the common notions of untaught Idomenians. So difficult it hath always been," says our author, "for great conquerors to know where to stop. "In the meantime, natural philosophy began to rise from its ashes, under the direction of a person of great genius, who is looked upon as having had something in him above Idomenian nature. He observed, that the Idomenian faculties were certainly intended for contemplation, and that the works of nature were a nobler subject to exercise them upon, than the follies of systems, or the errors of the learned; and being sensible of the difficulty of finding out the causes of natural things, he proposed, by accurate observation of the phænomena of nature, to find out the rules according to which they happen, without inquiring into the causes of those rules. In this he made considerable progress himself, and planned out much work for his followers, who call themselves inductive philosophers. The sceptics look with envy upon this rising sect, as eclipsing their reputation, and threatening to limit their empire; but they are at a loss on what hand to attack it. The vulgar begin to reverence it as producing useful discoveries. "It is to be observed, that every Idomenian firmly believes, that two or more bodies may exist in the same place. For this they have the testimony of sense, and they can no more doubt of it, than they can doubt whether they have any perception at all. They often see two bodies meet and coincide in the same place, and separate again, without having undergone any change in their sensible qualities by this To this quality of bodies they gave a name, which our author tells us hath no word answering to it in any human language. And, therefore, after making a long apology, which I omit, he begs leave to call it the overcoming quality of bodies. He assures us, that "the speculations which had been raised about this single quality of bodies, and the hypotheses contrived to account for it, were sufficient to fill many volumes. Nor have there been fewer hypotheses invented by their philosophers, to account for the changes of magnitude and figure; which, in most bodies that move, they perceive to be in a continual fluctuaation. The founder of the inductive sect, believing it to be above the reach of Idomenian faculties, to discover the real causes of these phænomena, applied himself to find from observation, by what laws they are connected together; and discovered many mathematical ratios and relations concerning the motions, magnitudes, figures, and overcoming quality of bodies, which constant experience confirms. But the opposers of this sect choose rather to content themselves with feigned causes of these phænomena, than to acknowledge the real laws whereby they are governed, which humble their pride, by being confessedly unaccountable." Thus far Johannes Rudolphus Anepigraphus. Whether this Anepigraphus be the same who is recorded among the Greek alchemistical writers not yet published, by Borrichius, Fabricius, and others, I do not pretend to determine. The identity of their name, and the similitude of their studies, although no slight arguments, yet are not absolutely conclusive. Nor will I take upon me to judge of the narrative of this learned traveler, by the external marks of his credibility; I shall confine myself to those which the crit.cs call internal. would even be of small importance to inquire, whether the Idomenians have a real, or only an ideal existence; since this is disputed among the learned with regard to things with which we are more nearly connected. The important question is, whether the account above given, is a just account of their geometry and philosophy? We have all the faculties which they It This is true; the name is not imaginary. "Anepigraphus the Philosopher" is the reputed author of several chemical treatises in Greek, which have not as yet been deemed worthy of publication. See Du Cange," Gloss, med. et inf., Graecitatis," voce Пents and Reinesii, "Var. Lectt." L. II. c. 5. -H. have, with the addition of others which they have not; we may, therefore, form some judgment of their philosophy and geometry, by separating from all others, the perceptions we have by sight and reasoning upon them. As far as I am able to judge in this way, after a careful examination, their geometry must be such as Anepigraphus hath described. Nor does his account of their philosophy appear to contain any evident marks of imposture; although here, no doubt, proper allowance is to be made for liberties which travellers take, as well as for involuntary mistakes which they are apt to fall into. Section X. OF THE PARALLEL MOTION OF THE EYES. Having explained, as distinctly as we can, visible figure, and shewn its connection with the things signified by it, it will be proper next to consider some phænomena of the eyes, and of vision, which have commonly been referred to custom, to anatomical or to mechanical causes; but which, as I conceive, must be resolved into original powers and principles of the human mind; and, therefore, belong properly to the subject of this inquiry. The first is the parallel motion of the eyes; by which, when one eye is turned to the right or to the left, upwards or downwards, or straight forwards, the other always goes along with it in the same direction. We see plainly, when both eyes are open, that they are always turned the same way, as if both were acted upon by the same motive force; and if one eye is shut, and the hand laid upon it, while the other turns various ways, we feel the eye that is shut turn at the same time, and that whether we will or not. What makes this phænomenon surprising is, that it is acknowledged, by all anatomists, that the muscles which move the two eyes, and the nerves which serve these muscles, are entirely distinct and unconnected. It would be thought very surprising and unaccountable to see a man, who, from his birth, never moved one arm, without moving the other precisely in the same manner, so as to keep them always parallel-yet it would not be more difficult to find the physical cause of such motion of the arms, than it is to find the cause of the parallel motion of the eyes, which is perfectly similar. The only cause that hath been assigned of this parallel motion of the eyes, is custom. We find by experience, it is said, when we begin to look at objects, that, in order to have distinct vision, it is necessary to turn both eyes the same way; therefore, we soon acquire the habit of doing it constantly, and by degrees lose the power of doing otherwise. This account of the matter seems to be insufficient; because habits are not got at once; it takes time to acquire and to confirm them; and if this motion of the eyes were got by habit, we should see children, when they are born, turn their eyes different ways, and move one without the other, as they do their hands or legs. I know some have affirmed that they are apt to do so. But I have never found it true from my own observation, although I have taken pains to make observations of this kind, and have had good opportunities. I have likewise consulted experienced midwives, mothers, and nurses, and found them agree, that they had never observed distortions of this kind in the eyes of children, but when they had reason to suspect convulsions, or some preternatural cause. It seems, therefore, to be extremely probable, that, previous to custom, there is something in the constitution, some natural instinct, which directs us to move both eyes always the same way.' We know not how the mind acts upon the body, nor by what power the muscles are contracted and relaxed-but we see that, in some of the voluntary, as well as in some of the involuntary motions, this power is so directed, that many muscles which have no material tie or connection,+ act in concert, each of them being taught to play its part in exact time and measure. Nor doth a company of expert players in a theatrical performance, or of excellent musicians in a concert, or of good dancers in a country dance, with more regularity and order, conspire and contribute their several parts, to produce one uniform effect, than a number of muscles do, in many of the animal functions, and in many voluntary actions. Yet we see such actions no less skilfully and regularly performed in children, and in those who know not that they have such muscles, than in the most skilful anatomist and physiologist. Who taught all the muscles that are concerned in sucking, in swallowing our food, in breathing, and in the several natural expulsions, to act their part in such regular order and exact measure? It was not custom surely. It was that same powerful and wise Being who made the fabric of the human body, and fixed the laws by which the mind operates upon every part |