text:
The Inner Life, v. I/Fifth Section/I

THE EGO AND THE PERSONALITY There are still many of our members who do not fully understand the problem of the higher and the lower self. Nor is this wonderful; for we are repeatedly told that there is only one consciousness, and yet we often clearly feel two, so it is not remarkable that students should be uncertain as to the real relation between these two, and should wonder whether the ego is entirely dissociated from his physical body and has an existence of his own among his fellows on his own plane. This problem of the lower and higher self is an old one, and it is undoubtedly difficult to realize that there is after all only one consciousness, and that the apparent difference is caused only by the limitations of the various vehicles. The whole consciousness works on its own higher mental plane, but in the case of the ordinary man only partially and vaguely as yet. So far as it is active it is always on the side of good, because it desires that which is favorable to its evolution as a soul. It puts a portion of itself down into lower matter, and that portion becomes so much more keenly and vividly conscious in that matter that it thinks and acts as though it were a separate being, forgetting its connection with that less developed yet far wider self-consciousness above. So sometimes it seems as though the fragment worked against the whole; but the man who is instructed declines to be deluded, and reaches back through the keen alert consciousness of the fragment to the true consciousness behind, which is as yet so little developed. Undoubtedly the ego is only very partially expressed by his physical body, yet we should not be accurate in speaking of him as dissociated from that body. If we figure the ego as a solid body and the physical plane as a surface, the solid body if laid upon that surface could manifest itself through that surface only as plane figure, which would obviously be an exceedingly partial expression. We can see also that if the various sides of the solid were laid upon the surface successively we might obtain expressions which differed considerably, though all of them would be imperfect, because in all cases the solid would have an extension in an entirely different direction, which could by no means be expressed in the two dimensions of the superficies. We shall obtain a nearly accurate symbolism of the facts as far as the ordinary man is concerned if we suppose the solid to be conscious only so far as it is in contact with the surface, although the results gained through the manifestation of such consciousness would inhere in the solid as a whole, and would be present in any later expression of it, even though that might differ considerably from previous expressions. It is only in the case of those already somewhat advanced that we can speak of the ego as having a conscious existence among his fellows on his own plane. From the moment that he breaks off from his group-soul and commences his separate existence, he is a conscious entity; but the consciousness is of an exceedingly vague nature. The only physical sensation which occasionally comes to some persons is at the moment of awakening in the morning. There is a state intermediate between sleeping and waking in which a man is blissfully conscious that he exists, and yet is not conscious of any surrounding objects, not capable of any moment. Indeed, he sometimes knows that any movement would break the spell of happiness and bring him down into the ordinary waking world, and so he endeavours to remain still as long as possible. That condition — a consciousness of existence and of intense bliss — closely resembles that of the ego of the average man upon the higher mental plane. He is wholly centred there only for the short time which intervenes between the end of one life in the heaven-world and the commencement of his next descent into incarnation; and during that short period there comes to him the flash of retrospect and prospect — a glimpse of what his last life has done for him, and of what his next life is intended to do. For many ages these glimpses are his only moments of full awakening, and it is his desire for a more perfect manifestation, his desire to feel himself more thoroughly alive and active, which drives him into the effort of incarnation. It is not desire for life in the ordinary sense of the word, but rather for that complete consciousness which involves the power to respond to all possible vibrations from the surroundings on every plane, so that he may attain the perfection of sympathy. When the ego is still undeveloped the forces of the higher mental plane pass through him practically without affecting him, as he cannot respond to more than a very few of these extremely fine vibrations. It needs powerful and comparatively coarse vibrations to affect him at first, and these do not exist upon his own plane, and for that reason he has to put himself down to lower levels in order to find them. Therefore full consciousness comes to him at first only in the lowest and densest of his vehicles, his attention being focussed for a long time down in the physical plane; so that, although that plane is so much lower than his own and offers so much less scope for activity, in those early stages he feels himself much more alive when he is working there. As the consciousness increases and widens its scope he gradually begins to work more and more in matter one stage higher — that is, in astral matter. At a much later stage, when he has attained to clear working in astral matter, he begins to be able also to express himself through the matter of his mental body and the end of his present effort is achieved when he works as fully and clearly in the matter of the causal body on the higher mental plane as he does now on the physical plane. These stages of full development of consciousness must not be confounded with the mere learning to use to some extent the respective vehicles. A man is using his astral body whenever he expresses an emotion; he is using his mental body whenever he thinks. But that is very far from his being able to utilize either of them as independent vehicles through which consciousness can be fully expressed. When a man is fully conscious in his astral body, he has already made a considerable amount of progress; when he has bridged over the chasm between the astral consciousness and the physical, day and night no longer exist for him, since he leads a life unbroken in its continuity. For him death also has ceased to exist, since he carries that unbroken consciousness not only through night and day, but also through the portals of death itself and up to the end of his life upon the astral plane. One step of further development lies open to him — the consciousness of the heaven-world; and then his life and memory are continuous during the whole of each descent into incarnation. Yet one step more raises the full consciousness to the level of the ego on the higher mental plane, and after that he has always with him the memory of all his lives, and he is capable of consciously directing the various lower manifestations of himself at all points of his progress. It must not be supposed that the development of any of these stages of consciousness is ever sudden. The actual rending of the veil between two stages is usually a fairly rapid process, sometimes even instantaneous. A man who has normally no memory of what happens on the astral plane may unintentionally, by some accident or illness, or intentionally by certain definite practices, bridge over the interval and make the connection, so that from that time onward his astral consciousness will be continuous, and his memory of what happens while the physical body is asleep will therefore be perfect. But long before such an effort or accident is possible for him the full consciousness must have been working in the astral body, even though in the physical life he knew nothing of it. In exactly the same way a man must have been for a long time thoroughly practised in the use of his mental body as a vehicle before he can hope to break the barrier between that and the astral, so that he can have the pleasure of continuous recollection. By analogy this lead us to see that the ego must have been fully conscious and active on his own plane for a long time before any knowledge of that existence can come through to us in our physical life. There are many in whom the ego has already to some extent awakened from the condition of mere bliss which was described above, and is at least partially conscious of his own surroundings, and therefore of other egos. From that time on he leads a life and has interests and activities on his own plane; but even then we must remember that he puts down into the personality only a very small part of himself, and that that part constantly becomes entangled in interests which, because they are so partial, are often along different lines from the general activities of the ego himself, who consequently does not pay any particular attention to the lower life of the personality, unless something rather unusual happens to it. When this stage is reached he usually comes under the influence of a Master; indeed often his first clear consciousness of anything outside himself is his touch with that Master. The tremendous power of the Master's influence magnetises him, draws his vibrations into harmony with its own, and multiplies manyfold the rate of his development. It rays upon him like sunshine upon a flower, and he evolves rapidly under its influence. This is why, while the earlier stages of progress are so slow as to be almost imperceptible, when the Master turns His attention upon the man, developes him and arouses his own will to take part in the work, the speed of his advancement increases in geometrical progression. Of that stream of divine influence poured upon the ego by the Master, the amount which can be passed on to the personality depends upon the connection between it and the ego, which is very different in different cases. There is almost infinite variety in human life. The spiritual force rays upon the ego, and some little of it certainly comes through into the personality, because though the ego has put forth a part of himself he does not cut himself off entirely from it, notwithstanding the fact that in the case of all ordinary people the ego and the personality are very different things. The ego in ordinary men has not much grasp of the personality, nor a clear conception of his purpose in sending it forth; and, again, the small piece which meets us in the personality grows to have ways and opinions of its own. It is developing by the experience which it gains, and this is passed on to the ego; but along with this real development it usually gathers a good deal which is hardly worthy of that name. It acquires knowledge, but also prejudices, which are not really knowledge at all. It does not become quite free from these prejudices — not only of knowledge (or rather its absence) but of feeling and action as well — until the man reaches adeptship. It gradually discovers these things to be prejudices, and progresses through them; but it has always a great deal of limitation from which the ego is entirely free. As to the amount of the spiritual force which is passed to the personality, one can only decide in a particular case by using clairvoyance. But something of it must flow through always, because the lower is attached to the higher, just as the hand is attached to the body by the arm. It is certain that the personality must get something, but it can have only what it has made itself able to receive. It is also a question of qualities. The Master often plays upon qualities in the ego which are much obscured in the personality, and in that case of course very little comes down. As only those experiences of the personality can be handed on to the spiritual or permanent ego which are compatible with his nature and interests, so only those impulses to which it is able to respond can express themselves in the personality. Remember, though, that the former tends to exclude the bad and the latter the good — or rather we should call them the material and the spiritual, for nothing is bad. One may sometimes see by clairvoyance many of these influences at work. On a certain day, for example, we may see a characteristic of the personality much intensified, with no outward reason. The cause is often to be found in what is taking place at some higher level — the stimulation of that quality in the ego. Sometimes a man finds himself overflowing with affection or devotion, and quite unable on the physical plane to understand why. The cause is usually, again, the stimulation of the ego, or it may be that the ego is taking some special interest in the personality for the time being. In meditation we sometimes draw such attention on the part of the ego, though it is well to keep in mind that we must try to reach up to join that higher activity, rather than to interrupt it to draw down its attention to the lower. The higher influence is certainly invited by right meditation, which is always effective, even though on the physical plane things may seem to be very dull and quite without zest. The reaching upwards of the ego himself often means his neglect to send energy down to the personality, and this, of course, leaves the latter feeling rather dull and in the shade. The extent, then, to which the personality is influenced by the effort of the Master depends upon two things principally — the strength of the connection at the time between the ego and the personality; and the particular work which the Master is doing upon the ego, that is, the particular qualities upon which He is playing. Meditation and the study of spiritual subjects in this earthly life make a very great difference in the life of the ego. The ordinary person who has not taken up spiritual matters seriously has only a thread of connection between the higher and the lower self. The personality in his case seems to be all, and the ego, though he undoubtedly exists on his own plane, is not at all likely to be doing anything actively there. He is very much like a chicken which is growing inside an egg. But in the case of some of us who have been making efforts in the right direction, we may hope that the ego is becoming quite vividly conscious. He has broken through his shell, and is living a life of great activity and power. As we go on, we shall become able to unify our personal consciousness with the life of the ego, as far as that is possible, and then we shall have only the one consciousness; even down here we shall have the consciousness of the ego, who will know all that is going on. But with many people at the present day there is often considerable opposition between the personality and the ego. There are other things to be taken into account. It is by no means always accurate to judge the ego by his manifestation in the personality. An ego of intensely practical type may make much more show on the physical plane than another of far higher development, if the energy of the latter happens to be concentrated almost exclusively upon the causal or buddhic levels. Therefore people who see only on the physical plane are frequently entirely wrong in their estimation of the relative position of others. If you have to deal with a fairly advanced ego, you will sometimes find him rather inconsiderate of his body. You see whatever is put down into the personality is so much taken from him! I have again and again seen cases in which the ego was to some extent impatient and withdrew into himself somewhat; but on the other hand in cases such as these there is always a flow between the ego and the personality, which is not possible with the ordinary man. In the ordinary man the part is, as it were, put down and left, though not of course quite cut off; but at this more advanced stage there is a constant communication between the two along the channel. Therefore, the ego can withdraw a great deal of himself whenever he chooses, and leave a very poor representation of the real man behind. So the relation between the lower and the higher self varies much in different people and at different stages of development. As to the work of the ego, he may be learning things on his own plane; or he may be helping other egos — there are many kinds of work for which he may need an accession of strength. And then he may forget for a time to pay his personality proper attention, just as even a good man may occasionally, under some special pressure of business, forget his horse or his dog. Sometimes when that happens the personality reminds him of its existence by blundering into some foolishness which causes serious suffering. You may have noticed that sometimes, after you have completed a special piece of work that has needed the co-operation of the ego to a large extent — as, for example, lecturing to a large audience — he takes away the energy and leaves the personality with only enough to feel rather dispirited. For a time he admitted that there was some importance in the work, and therefore poured down a little more of himself, but afterwards he leaves the poor personality feeling rather depressed. Of course, depression comes much more often from other reasons, such as the presence of an astral entity in a low-spirited condition, or of some non-human beings. And joy also is not always due to the influence of the ego, for the fact is that the man does not think much about his own feelings when he is in a fit condition to receive an influx of power. Joy may be produced by the proximity of harmonious nature-spirits, or in a variety of other ways. The channel between the ego and personality is by no means always open. Sometimes it appears to be almost choked up — a condition of affairs which is quite a possibility in view of its narrowness in most cases. Then the force may break through again on some occasion, such as that of a conversion. But for many of us there is a constant flow in some measure. Meditation, conscientiously done, opens the channel and keeps it open. Always remember, though, that it is better to try to go up to the ego than to bring it down to the personality. Every ego has a certain knowledge of his own. He obtains a glimpse, between lives, of his past and future; in the undeveloped man this awakens the ego for a moment, after which he falls asleep again. During physical life the ordinary ego is to some extent capable of brooding watchfulness and a little effort, but is still in a sleepy condition. With a developed man the ego is fully awake. The ego in course of time discovers that there are a good many things which he can do, and when this happens he may rise into a condition in which he has a definite life on his own plane, though in many cases it is even then but dreamy. It is the ego's purpose to learn to be fully active on all planes, even the physical. Suppose you have an ego whose principal method of manifesting himself is by affection. That quality is what he wants exhibited by his personality, and if you down here try to feel strong affection and make a specialty of that, the ego will promptly throw more of himself down into the personality, because he finds in it exactly what he desires. Be careful to provide what he needs, and he will quickly take advantage of it. Egos on their own plane can help other egos, when they are sufficiently developed to do so. The ego of the ordinary person has rather a vegetable consciousness or life, and seems to be only just aware of other egos. The personality will not know what the ego does, unless they have been unified. The ego may know the Master while the personality does not. The study of inner things, and living the life, wakes up the ego. Purely unselfish devotion belongs to the higher planes and concerns him. I do not think the experiences of the personality can be transmitted to the ego, but the essence of them may. He cares little for the details, but he wants the essence of it. Any of those thoughts that we consider evil are impossible for the ego. For precise definition he must come down into the physical body. He devotes himself more especially during the heaven-life to the assimilation of the experiences of the personality, but he is doing it all the time. When you take up the study of Theosophy, and live the life, you begin to call the attention of the ego by sending up vibrations to which he can respond. The ordinary man has in his life little that appeals to the ego. High unselfish affection and devotion belong to the highest astral sub-plane, and these reflect themselves in the corresponding matter of the mental plane, so that they touch the causal, not the lower mental. Thus only unselfish thoughts affect the ego. All the lower thoughts affect the permanent atoms, but not the ego; and corresponding to them you would find gaps in the causal body, not bad colors. Selfishness below shows in it as absence of affection or sympathy, and when the good quality developes the gap will be filled up. In the causal body you can see whether a man can possibly fail in this or that quality. Try to deve1ope the qualities the ego wants, and he will come down to help. As is said in Light on the Path, watch for the ego, and let him fight through you, and yet at the same time never forget that you are the ego. Therefore identify yourself with him and make the lower give way to you the higher. Yet do not be too greatly disheartened if you should fall even many times, for even failure is to a certain extent a success, since we learn by it and so are wiser to meet the next problem. We cannot always succeed now at every point, though we surely shall do so ultimately. But never forget that it is not expected of us that we shall always succeed, but only that we shall do our best.

title: The Inner Life, v. I/Fifth Section/I


text:
The Inner Life, v. I/Fifth Section/II

COUNTERPARTS When the ego descends into incarnation, he draws round himself a mass of astral matter, not yet formed into a definite astral body; this takes, in the first place, the shape of that ovoid which is the nearest expression that we can realize of the true shape of the causal body. But when the further step downward and outward into physical incarnation is taken, and a little physical body is formed in the midst of that astral matter, it immediately begins to exert a violent attraction over it, so that the great majority of the astral matter (which previously may be thought of as fairly evenly distributed over the large oval) now becomes concentrated into the periphery of that physical body. As the physical body grows, the astral matter follows every change, and thus we find man presenting the spectacle of an astral body, ninety-nine percent of which is compressed within the periphery of his physical body, only about the remaining one percent filling the rest of the ovoid form. In the plates in Man, Visible and Invisible we have sketched in the outline of the physical body merely in pencil so that it shows but slightly, because my especial desire in that book was to emphasize the colors of the ovoid, and the way in which they illustrate the development of man by the transfer of vibrations from the lower bodies to the higher; but in reality that astral counterpart of the physical body is very solid and definite, and quite clearly distinguishable from the surrounding ovoid. Note, therefore, that the astral matter takes the exact form of the physical matter merely because of the attraction which the latter has for the former. But we must further realize that although we may speak of the lowest sub-plane of the astral as corresponding to solid physical matter, it is yet very different in texture, for all astral matter bears to its corresponding physical matter something the same sort of relation that the liquid bears to the solid. Therefore the particles of the astral body, whether in the finest or coarsest parts of it, are constantly in motion among themselves, just as are particles of flowing water; and it will consequently be seen that it is quite impossible for the astral body to possess specialized organs in the same sense as does the physical body. No doubt there is an exact counterpart in astral matter of the rods and cones which make up the retina of the physical eye; but the particles which at one moment are occupying that particular position in an astral body may, a second or two later, be moving through the hand or the foot. One does not, therefore, see upon the astral plane by means of the astral counterpart of the physical eyes, nor does one hear with the astral counterpart of the physical ears; indeed, it is perhaps not exactly correct to apply the terms “seeing” and “hearing” to astral methods of cognizance, since these terms are commonly held to imply specialized sense-organs, whereas the fact is that every particle in the astral body is capable of receiving and transmitting vibrations from one of its own type, but its own type only. Thus when one obtains a glimpse of astral consciousness, one is surprised to find oneself able to see on all sides simultaneously, instead of only in front as one does on the physical plane. The exact correspondence of the astral body to the physical therefore is merely a matter of external form, and does not at all involve any similarity of function in the various organs. But the attraction continued all through life sets up a kind of habit or momentum in the astral matter, which causes it to retain the same form even while it is withdrawn temporarily from the attraction of the physical body at night and permanently after death; so that even through the long astral life the lineaments of the physical body which was put aside at death will still be preserved almost unchanged. Almost — because we must not forget that thought has a powerful influence upon astral matter and can readily mould it, so that a man who habitually thinks of himself after death as younger than he actually was at the time of that death will gradually come to present a somewhat younger appearance. A questioner asks, “If the arm of a man, the branch of a tree, or the leg of a chair were cut off, would in each case the astral counterpart also be removed, and can we, by breaking an astral counterpart, produce a fracture in a physical object? That is to say, if with the hand of my astral body I break the astral counterpart of a chair, will the physical chair also be broke?” The three cases given are not quite analogous. Both the tree and the man have the life within them which makes the astral body in each case a coherent whole. It is strongly attracted by the particles of the physical body, and therefore adapts itself to its shape, but if part of that physical body be removed, the coherence of the living astral matter is stronger than the attraction towards that severed portion of the physical. Consequently the astral counterpart of the arm or branch will not be carried away with the severed physical fragment. Since it has acquired the habit of keeping that particular form, it will continue for a short time to retain the original shape, but will soon withdraw within the limits of the maimed form.

In the case of an inanimate body, such as a chair or a basin, there would not be the same kind of individual life to maintain cohesion. Consequently when the physical object was broken the astral counterpart would also be divided; but it would not be possible to break an astral counterpart, and in that way to affect the physical object. In other words the act of fraction must begin on the physical plane. One could of course move a purely astral object by means of an astral hand if one wished, but not the astral counterpart of a physical object. In order to perform this latter feat it would be necessary to materialize a hand and move the physical object, when the astral counterpart would of course accompany it. The astral counterpart is there because the physical object is there, just as the scent of a rose fills the room because the rose is there. To suggest that by moving the astral counterpart one could also move the physical object is like suggesting that by moving the smell one could move the physical rose which causes the smell. The astral body changes its particles as does the physical, but fortunately the clumsy and tiresome process of cooking, eating and digesting food is not a necessity on the astral plane. The particles which fall away are replaced by others from the surrounding atmosphere. The purely physical cravings of hunger and thirst no longer exist there; but the desire of the glutton to gratify the sensation of taste, and the desire of the drunkard for the exhilaration which follows, for him, the absorption of alcohol — these are both astral, and therefore they still persist, and cause great suffering because of the absence of the physical body through which alone they could be satisfied. So far as we are at present aware the astral body does not appear to be susceptible to fatigue. The ordinary man while possessing a physical body naturally never has the opportunity of working for any length of time consecutively upon the astral plane, for his nights of astral work alternate with days of physical work. I knew, however, of one case of a man who, having the right to take a rapid reincarnation, had to wait upon the astral plane twenty-five years for the special conditions which he required. He spent the whole of this time in working for the help of others, without any intermission except the occasional attendance at classes held by pupils of our Masters; and he assured me that he had never felt the slightest sense of fatigue — that in fact he had forgotten what it meant to be tired. We all know that excessive or long-continued emotion tires us very quickly in ordinary life, and since emotion is an expression of the astral, that may perhaps lead some to suppose that fatigue of the astral body is possible. I think, however, that it will be found that what is subject to fatigue is merely the physical organism through which everything in us which manifests on this plane must pass. What we call mental fatigue is a parallel case. There is no such thing as fatigue in the mind; what we call by that name is only fatigue of the physical brain through which that mind has to express itself. A spectator who has not been able to raise his sight above the astral level will of course see only astral matter when he looks at the aura of his fellow-men. He will see that this astral matter not only surrounds the physical body but also interpenetrates it, and that within the periphery of that body it is much more densely aggregated than in that part of the aura which lies outside. This is due to the attraction of the large amount of dense astral matter which is gathered together there as the counterpart of the cells of the physical body. When during sleep the astral body is drawn from the physical this arrangement still persists, and then any one looking at the astral body with clairvoyant vision would still see, just as before, a form resembling the physical body surrounded by an aura. That form would now be composed only of astral matter, but still the great difference in density between it and its surrounding mist would be quite sufficient to make it clearly distinguishable, even though it is itself only a form of denser mist. There is a considerable difference in appearance between the evolved and the unevolved man. Even in the case of the latter the features and shape of the inner form are recognizable always, though blurred and indistinct; but the surrounding egg scarcely deserves the name for it is in fact a mere shapeless wreath of mist, having neither regularity nor permanence of outline. In the more developed man the change is very marked, both in the aura and the form within it. This latter is much more distinct and definite — a closer reproduction of the man's physical appearance; while instead of the floating mist-wreath we see a sharply defined ovoid form preserving its shape unaffected amidst all the varied currents which arc always swirling round it on the astral plane. Though the arrangement of the astral body is largely changed after death by the action of the desire elemental, such alteration does not in any way affect the recognizability of the form within the egg, though the natural changes which take place tend on the whole to make the form grow somewhat fainter and more spiritual in appearance as time passes on.

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title: The Inner Life, v. I/Fifth Section/II


text:
The Inner Life, v. I/Fifth Section/III

COLORS IN THE ASTRAL BODY Any comparatively permanent color in the astral body means a persistent vibration, which in the course of time produces its effect upon the mental body, and also upon the causal body, so that the higher qualities developed by the life on the lower planes are gradually built into the permanent causal body, and so become qualities of the soul itself. The colors may be mingled to any extent; for example, affection (rose) mingled with religious devotion (blue) will give a lovely violet. It is only the good thought or feeling which can produce an effect in the causal body, and so be permanently stored up as part of the man. Other kinds of thought and feeling remain in the lower vehicles and are comparatively impermanent. The size a thought-form shows the strength of the emotion.

title: The Inner Life, v. I/Fifth Section/III


text:
The Inner Life, v. I/Fifth Section/IV

THE CAUSAL BODY No number of physical bodies could fully contain the causal body, any more than any number of lines can make a square, or any number of squares can make a cube. The ego puts himself down into his various bodies with the hope of gaining two things — to make the causal body learn to respond to more vibrations, and also to increase its size. Most people are not more than just conscious in the causal body. The strings of such egos cannot be played upon directly, but are affected from below by way of overtones. Most men can at present only work on the matter of the third sub-plane of the mental (the lowest part of their causal bodies), and indeed only the lower matter even of that is usually in operation. When they are on the Path, the second sub-plane opens up. The adept uses the whole causal body while his consciousness is on the physical plane. A rough and ready way of deciding at what stage a man stands is to look at the causal body. It shows also how he arrived there. Men develop unequally — we are all undeveloped in some way. An animal has a minimum-sized causal body as soon as he is individualized; then it has to be developed both as to size and color.

title: The Inner Life, v. I/Fifth Section/IV


text:
The Mathematical Principles of Natural Philosophy (1729)/Book 2/Section 2

If a body is resisted in the duplicate ratio of its velocity, and moves by its innate force only through a similar medium; and the times be taken in a geometrical progression, proceeding from less to greater terms: I say that the velocities at the beginning of each of the times are in the same geometrical progression inversely; and that the spaces are equal, which are described in each of the times.

For since the resistance of the medium is proportional to the square of the velocity, and the decrement of the velocity is proportional to the resistance; if the time be divided into innumerable equal particles, the squares of the velocities at the beginning of each of the times will be proportional to the differences of the same velocities. Let those particles of time be AK, KL, LM, &c. taken in the right line CD; and erect the perpendiculars AB, Kk, Ll, Mm, &c. meeting the Hyperbola BklmG, described with the centre C and the rectangular asymptotes CD, CH, in B, k, l, m, &c. then AB will be to Kk as CK to CA, and, by division, AB - Kk to Kk as AK to CA, and, alternately, AB - Kk to AK as Kk to CA, and therefore as AB x Kk to AB x CA. Therefore since AK and AB x CA are given, AB - Kk will be as AB x Kk; and lastly, when AB and Kk coincide, as AB ². And, by the like reasoning, Kk - Ll, Ll - Mm, &c. will be as Kk², Ll², &c. Therefore the squares of the lines AB, Kk, Ll, Mm, &c. are as their differences; and therefore, since the squares of the velocities were shewn above to be as their differences, the progression of both will be alike. This being demonstrated, it follows also that the areas described by these lines are in a like progression with the spaces described by these velocities. Therefore if the velocity at the beginning of the first time AK be expounded by the line AB, and the velocity at the beginning of the second time KL by the line Kk, and the length described in the first time by the area AKkB; all the following velocities will be expounded by the following lines Ll, Mm, &c. and the lengths described, by the areas Kl, Lm, &c. And, by composition, if the whole time be expounded by AM, the sum of its parts, the whole length described will be expounded by AMmB the sum of its parts. Now conceive the time AM to be divided into the parts AK, KL, LM, &c. so that CA, CK, CL, CM, &c. may be in a geometrical progression; and those parts will be in the same progression, and the velocities AB, Kk, Ll, Mm, &c. will be in the same progression inversly, and the spaces described Ak, Kl, Lm, &c. will be equal. Q.E.D.

Corollary 1. Hence it appears, that if the time be expounded by any part AD of the asymptote, and the velocity in the beginning of the time by the ordinate AB; the velocity at the end of the time will be expounded by the ordinate DG; and the whole space described, by the adjacent hyperbolic area ABCD; and the space which any body can describe in the same time AD, with the first velocity AB, in a nonresisting medium, by the rectangle AB x AD.

Corollary 2. Hence the space described in a resisting medium is given, by taking it to the space described with the uniform velocity AB in a non-resisting medium, as the hyperbolic area ABGD to the rectangle AB x AD.

Corollary 3. The resistance of the medium is also given, by making it equal, in the very beginning of the motion, to an uniform centripetal force, which could generate, in a body falling thro' a non-resisting medium, the velocity AB, in the time AC. For if BT be drawn touching the hyperbola in B, and meeting the asymptote in T; the right line AT will be equal to AC, and will express the time, in which the first resistance uniformly continued, may take away the whole velocity AB.

Corollary 4. And thence is also given the proportion of this resistance to the force of gravity, or any other given centripetal force.

Corollary 5. And vice versa, if there is given the proportion of the resistance to any given centripetal force; the time AC is also given, in which a centripetal force equal to the resistance may generate any velocity as AB; and thence is given the point B, through which the hyperbola, having CH, CD for its asymptotes, is to be described; as also the space ABGD, which a body, by beginning its motion with that velocity AB, can describe in any time AD, in a similar resisting medium.

Homogeneous and equal spherical bodies, oppos'd by resistances that are in the duplicate ratio of the velocities, and moving on by their innate force only, will, in times which are reciprocally as the velocities at the beginning, describe equal spaces, and lose parts of their velocities proportional to the wholes.

To the rectangular asymptotes CD, CH describe any hyperbola BbEe, cutting the perpendiculars AB, ab, DE, de, in B, b, E, e; let the initial velocities be expounded by the perpendiculars AB, DE, and the times by the lines Aa, Dd. Therefore as Aa is to Dd, so (by the hypothesis) is DE to AB, and so (from the nature of the hyperbola) is CA to CD; and, by composition, so is Ca to Cd. Therefore the areas ABba, DEed, that is, the spaces described, are equal among themselves, and the first velocities AB, DE are proportional to the last ab, de; and therefore, by division, proportional to the parts of the velocities lost, AB - ab, DE - de. Q.E.D.

If spherical bodies are resisted in the duplicate ratio of their velocities, in times which are as the first motions directly and the first resistances inversely, they will lose parts of their motions proportional to the wholes, and will describe spaces proportional to those times and the first velocities conjunctly.

For the parts of the motions lost are as the resistances and times conjunctly. Therefore, that those parts may be be proportional to the wholes, the resistance and time conjunctly ought to be as the motion. Therefore the time will be as the motion directly and the resistance inversely. Wherefore the particles of the times being taken in that ratio, the bodies will always lose parts of their motions proportional to the wholes, and therefore will retain velocities always proportional to their first velocities. And because of the given ratio of the velocities, they will always describe spaces, which are as the first velocities and the times conjunctly. Q.E.D.

Corollary 1. Therefore if bodies equally swift are resisted in a duplicate ratio of their diameters: Homogeneous globes moving with any velocities whatsoever, by describing spaces proportional to their diameters, will lose parts of their motions proportional to the wholes. For the motion of each globe will be as its velocity and mass conjunctly, that is, as the velocity and the cube of its diameter; the resistance (by supposition) will be as the square of the diameter and the square of the velocity conjunctly; and the time (by this proposition) is in the former ratio directly and in the latter inversely, that is, as the diameter directly and the velocity inversely; and therefore the space, which is proportional to the time and velocity, is as the diameter.

Corollary 2. If bodies equally swift are resisted in a sesquiplicate ratio of their diameters: Homogeneous globes, moving with any velocities whatsoever, by describing spaces that are in a sesquiplicate ratio of the diameters, will lose parts of their motions proportional to the wholes.

Corollary 3. And universally, if equally swift bodies are resisted in the ratio of any power of the diameters: the spaces, in which homogeneous globes, moving with any velocity whatsoever, will lose parts of their motions proportional to the wholes, will be as the cubes of the diameters applied to that power. Let those diameters applied to that power. Let those diameters be D and E; and if the resistances, where the velocities are supposed equal, are as D" and E": the spaces in which the globes, moving with any velocities whatsoever, will lose parts of their motions proportional to the wholes, will be as D3-"and E3-". And therefore homogeneous globes, in describing spaces proportional to D3-"and E3-" will retain their velocities in the same ratio to one another as at the beginning.

Corollary 4. Now if the globes are not homogeneous, the space described by the denser globe must be augmented in the ratio of the density. For the motion, with an equal velocity, is greater in the ratio of the density, and the time (by this Proposition) is augmented in the ratio of motion directly, and the space described in the ratio of the time.

Corollary 5. And if the globes move in different mediums, the space, in a medium which, caeteris paribus, resists the most, must be diminished in the ratio of the greater resistance. For the time (by this proposition) will be diminished in the ratio of the augmented resistance, and the space in the ratio of the time.

Lemma II. The moment of any Genitum is equal to the moments of each of the generating sides drawn into the indices of the powers of those sides, and into their coefficients continually.

I call any quantity a Genitum, which is not made addition or subduction of divers parts, but is generated or produced in arithmetic by the multiplication, division, or extraction of the root of any terms whatsoever; in geometry by the invention of contents and sides, or of the extreams and means of proportionals, Quantities of this kind are products, quotients, roots, rectangles, squares, cubes, square and cubic sides, and the like. These quantities I here consider as variable and indetermined, and increasing or decreasing as it were by a perpetual motion or flux; and I understand their momentaneous increments or decrements by the name of Moments; so that the increments may be esteem'd as added, or affirmative moments; and the decrements as subducted, or negative ones. But take care not to look upon finite particles as such. Finite particles are not moments, but the very quantities generated by the moments. We are to conceive them as the just nascent principles of finite magnitudes. Nor do we in this Lemma regard the magnitude of the moments, but their first proportion as nascent. It will be the same thing, if, instead of moments, we use either the Velocities of the increments and decrements (which may also be called the motions, mutations, and fluxions of quantities) or any finite quantities proportional to those velocities. The coefficient of any generating side is the quantity which arises by applying the Genitum to that side.

Wherefore the sense of the Lemma is, that if the moments of any quantities A, B, C, &c. increasing or decreasing by a perpetual flux, or the velocities of the mutations which are proportional to them, be called a, b, c, &c. the moment or mutation of the generated rectangle AB will be aB + bA; the moment of the generated content ABC will be aBC + bAC + cAB: and the moments of the generated powers, A², A³, A4, A1/2, A3/2, A1/3, A2/3, A-1, A-2 , A-1/2 will be 2aA, 3aA², 4aA³, 1/2aA-3/2, 3/2aA1/2, 1/3aA-2/3, 2/3aA-1/3, -aA-2 , -2aA-3, -½aA-3/2 respectively. And in general, that the moment of any power An/m will be n/mA(n-m)/m. Also that the moment of the generated quantity A².B will be 2aAB + bA²; the moment os the generated quantity A³.B4.C² will be 3aA².B4.C² + 4bA³B³C² + 2cA³B4.C; and the moment of the generated quantity A³/B² or A³.B-2 will be 3aA²B-2 - 2bA³B-3; and so on. The Lemma is thus demonstrated.

Case 1. Any rectangle as AB augmented by a perpetual flux, when, as yet, there wanted of the sides A and B half their moments ½a and ½b, was A - ½a into B - ½b or AB + ½aB + ½bA + ½ab; but as soon as the sides A and B are augmented by the other half moments; the rectangle becomes A + ½a into B + ½b or AB + ½aB + ½bA + ½ab. Fr0m this rectangle subduct the former rectangle, and there will remain the excess aB + bA. Therefore with the whole increments a and b of the sides, the increment aB + bA of the rectangle is generated. Q.E.D.

Case 2. Suppose AB always equal to G, and then the moment of the content ABC or GC (by case 1.) will be gC + cG, that is, (putting AB and aB + bA for G and g) aBC + bAC + cAB. And the reasoning is the same for contents under never so many sides. Q.E.D.

Case 3. Suppose the sides A, B, and C, to be always equal among themselves; and the moment aB + bA, of A², that is, of the rectangle AB, will be 2aA; and the moment aBC + bAC + cAB of A³, that is, of the content ABC, will be 3aA². And by the same reasoning the moment of any power An is naAn-1. Q.E.D.

Case 4. Therefore since 1/A into A is 1, the moment of 1/A drawn into A, together with 1/A drawn into a, will be the moment of 1, that is, nothing. Therefore the moment of 1/A or of A-1 is -a/A². And generally, since 1/An into An is 1, the moment of 1/An drawn into An together with 1/An into naA1/An-1 will be nothing. And therefore the moment of 1/An or A-n will be -na/An+1. Q.E.D.

Case 5. And since A1/2 into A1/2 is A, the moment of A1/2 drawn into 2A1/2 will be a, (by case 3:) and therefore the moment of A1/22 will be a/2A1/2 or ½aA-1/2. And generally, putting Am/n equal to B, then Am will be equal to Bn, and therefore maAm-1 equal to nbBn-1, and maA-1 equal to nbB-1 or nbA-m/n; and therefore m/naA(m-n)/n is equal to b, that is, equal to the moment of Am/n. Q.E.D.

Case 6. Therefore the moment of any generated quantity AmBn is the moment of Am drawn into Bn, together with the moment of Bn drawn into Am, that is, maAm-1Bn + nbBn-1Am; and that whether the indices m and n of the powers be whole numbers or fractions, affirmative or negative. And the reasoning is the same for contents under more powers. Q.E.D.

Corollary 1. Hence in quantities continually proportional, if one term is given, the moments of the rest of the terms will be as the same terms multiplied by the number of intervals between them and the given term. Let A, B, C, D, E, F, be continually proportional; then if the term C is given, the moments of the rest of the terms will be among themselves, as -2A, -B, D, 2E, 3F.

Corollary 2. And if in four proportionals the two means are given, the moments of the extremes will be as those extremes. The same is to be understood of the sides of any given rectangle.

Corollary 3. And if the sum or difference os two squares is given, the moments of the sides will be reciprocally as the sides.

In a letter of mine to Mr. J. Collins, dated December 10. 1572. having described a method of Tangents, which I suspected to be the same with Slusin's method, which at that time was not made publick; I subjoined these words; This is one particular, or rather a corollary, of a general method, which extends itself, without any troublesome calculation, not only to the drawing of Tangents to any Curve lines, whether Geometrical or Mechanicals or any how respecting right lines or other Curves, bttt also to the resolving other abstruser kinds of Problems about the crookedness, areas, lengths centres of gravity of Curves, &c. nor is it (as Hudden's method de Maximis & Minimis) limited to equations which are free from surd quantities. This method I have interwoven with that other of working in equations, by reducing them to infinite series. So far that letter. And these last words relate to a Treatise I composed on that subject in the year 1671. The foundation of that general method is contained in the preceding Lemma.

If a body in an uniform medium, being uniformly acted upon by the force of gravity, ascends or descends in a right line; and the whole space described be distinguished into equal parts, and in the beginning of each of the parts, (by adding or subducting the resisting force of the medium to or from the force of gravity, when the body ascends or descends) you collect the absolute forces; I say that those absolute forces are in a geometrical progression. Plate 2. Figure 1.

For let the force of gravity be expounded by the given line AC; the force of resistance by the indefinite line AK; the absolute force in the descent of the body, by the difference KC; the velocity os the body by a line AP, which shall be a mean proportional between AK and AC, and therefore in a subduplicate ratio of the resistance; the increment of the resistance made in a given particle of time by the lineola KL, and the contemporaneous increment of the velocity by the lineola PQ; and with the centre C, and rectangular asymptotes CA, CH, describe any Hyperbola BNS, meeting the erected perpendiculars AB, KN, LO in B, N, and O. Because AK is as AP², the moment KL of the one will be as the moment 2APQ of the other, that is, as AP * KC, for the increment PO of the velocity is (by law 2.) proportional to the generating force KC. Let the ratio of KL be compounded with the ratio of KN, and the rectangle KL * KN will become as AP * KC * KN; that is, (because the rectangle KC * KN is given) as AP. But the ultimate ratio of the hyperbolic area KNOL to the rectangle KL * KN becomes, when the points K and L coincide, the ratio of equality. Therefore that hyperbolic evanescent area is as AP. Therefore the whole hyperbolic area ABOL is composed of particles KNOL which are always proportional to the velocity AP; and therefore is itself proportional to the space described with that velocity. Let that area be now divided into equal parts, as ABMI, IMNK, KNOL, &c. and the absolute forces AC, IC, KC, LC, &c. will be in a geometrical progression. Q.E.D. And by a like reasoning, in the ascent of the body, taking, on the contrary side of the point A, the equal areas ABmi, imnk, knol, &c. it will appear that the absolute forces AC, iC, kC, lC, &c. are continually proportional. Therefore if all the spaces in the ascent and descent are taken equal; all the absolute forces lC, kC, IC, iC, AC, IC, KC, LC, &c. will be continually proportional. Q.E.D.

Corollary 1. Hence if the space described be expounded by the hyperbolic area ABNK; the force of gravity, the velocity of the body, and the resistance of the medium, may be expounded by the lines AC, AP, and AK respectively; and vice versa.

Corollary 2. And the greatest velocity, which the body can ever acquire in an infinite descent, will be expounded by the line AC.

Corollary 3. Therefore if the resistance of the medium answering to any given velocity be known, the greatest velocity will be found, by taking it to that given velocity in a ratio subduplicate of the ratio which the force of gravity bears to that known resistance of the medium.

Supposing what is above demonstrated, I say that if the tangents of the angles of the sector of a circle, and of an hyperbola, be taken proportional to the velocities, the radius being of a fit magnitude; all the time of the ascent to the highest place will be as the sector of the circle, and all the time of descending from the highest place as the sector of the hyperbola. Plate 2. Figure 2.

To the right line AC, which expresses the force of gravity, let AD be drawn perpendicular and equal. From the centre D with the semidiameter AD describe as well the quadrant AtE of a Circle; as the rectangular Hyperbola AVZ, whose axe is AX, principal vertex A, and asymptote DC. Let Dp, DP be drawn; and the circular sector AtD will be as all the time of the ascent to the highest place; and the hyperbolic sector ATD as all the time of descent from the highest place: If so be that the tangents Ap, AP of those sectors be as the velocities. (Figure 2).

Case 1. Draw Dvq cutting off the moments or least particles tDv and qDp, described in the same time, of the lector ADt and of the triangle ADp. Since those particles (because of the common angle D) are in a duplicate ratio os the sides, the particle tDv will be as qDp * tD² / pD², that is, (because tD is given) as qDp / pD². But pD² is AD² + Ap², that is, AD² + AD * Ak, or AD * Ck; and qDp is ½AD * pq. Therefore: Dv, the particle of the sector, is as pq/Ck; that is, as the least decrement pq of the velocity directly, and the force Ck, which diminishes the velocity, inversely; and therefore as the particle of time answering to the decrement of the velocity. And, by composition, the sum of all the particles tDv in the sector ADtt will be as the sum of the particles of time answering to each of the lost particles pq, of the decreasing velocity Ap, till that velocity, being diminished into nothing, vanishes; that is, the whole sector ADt is as the whole time of ascent to the highest place. Q.E.D.

Case 2. Draw DQV cutting off the least particles TDV and PDQ of the sector DAV, and of the triangle DAQ; and these particles will be to each other as DT² to DP², that is, (if TX and AP are parallel) as DX² to DA² or TX² to AP²; and, by division, as DX² - TX² to DA² - AP². But, from the nature of the hyperbola, DX² - TX² is AD²; and, by the supposition, AP² is AD * AK. Therefore the particles are to each other as AD² to AD² - AD * AK; that is, as AD to AD - AK or AC to CK: and therefore the particle TDV of the sector is PDQ * AC / CK; and therefore (because AC and AD are given) as PQ / CK that is, as the increment of the velocity directly, and as the force generating the increment inversely; and therefore as the particle of the time answering to the increment. And, by composition, the sum of the particles of time, in which all the particles PQ of the velocity AP are generated, will be as the sum of the particles of the sector ATD; that is, the whole time will be as the whole sector. Q.E.D.

Corollary 1. Hence is AB be equal to a fourth part of AC, the space which a body will describe by falling in any time will be to the space which the body could describe, by moving uniformly on in the same time with its greatest velocity AC, as the area ABNK, which expresses the space described in falling, to the area ATD, which expresses the time. For since AC is to AP as AP to AK, then (by corollary 1. lemma 2. of this Book) LK is to PQ as 2AK to AP, that is, as 2AP to AC, and thence LK is to ½PQ as AP to ¼AC or AB; and KN is to AC or AD as AB to CK; and therefore, ex œquo, LKNO is to DTV as CK to ACC. Therefore, ex æquo, LKNO is to DTV as AP to AC; that is, as the velocity of the falling body to the greatest velocity which the body by falling can acquire. Since therefore the moments LKNO and DTV of the areas ABNK and ATD are as the velocities, all the parts of those areas generated in the same time, will be as the spaces described in the same time; and therefore the whole areas ABNK and ADT generated from the beginning, will be as the whole spaces described from the beginning of the descent. Q.E.D.

Corollary 2. The same is true also of the space described in the ascent. That is to say, that all that space is to the space described in the same time with the uniform velocity AC, as the area ABnk is to the sector ADt.

Corollary 3. The velocity of the body, falling in the time ATD, is to the velocity which it would acquire in the same time in a non-resisting space, as the triangle APD to the hyperbolic sector ATD. For the velocity in a non-resisting medium would be as the time ATD, and in a resisting medium is as AP, that is, as the triangle APD. And those velocities at the beginning of the descent, are equal among themselves, as well as those areas ATD, APD.

Corollary 4. By the same argument, the velocity in the ascent is to the velocity with which the body in the same time, in a non-resisting space, would lose all its motion of ascent, as the triangle ApD to the circular sector AtD; or as the right line Ap to the arc At.

Corollary 5. Therefore rhe time in which a body by falling in a resisting medium, would acquire the velocity AP, is to the time in which it would acquire its greatest velocity AC by falling in a non-resisting space, as the sector ADT to the triangle ADC: and the time in which it would lose its velocity Ap by ascending in a resisting medium, is to the time in which it would lose the same velocity by ascending in a nonresisting space, as the arc At to its tangent Ap.

Corollary 6. Hence from the given time there is given the space described in the ascent or descent. For the greatest velocity of a body descending in infinitum is given (by Corollory 2 and 3. Theorem 6. of this Book) and thence the time is given in which a body would acquire that velocity by falling in a non-resisting space. And taking the sector ADT or ADt to the triangle ADC in the ratio of the given time to the time just now found; there will be given both the velocity AP or Ap, and the area ABNK or ABnk, which is to the sector ADT, or ADt, as the space sought to the space which would, in the given time, be uniformly described with that greatest velocity sound just before.

Corollary 7. And by going backward, from the given space of ascent or descent ABnk or ABNK, there will be given the time ADt or ADT.

Suppose the uniform force of gravity to tend directly to the plane of the horizon, and the resistance to be as the density of the medium and the square of the velocity conjunctly: it is proposed to find the density of the medium in each place, which shall make the body move in any given curve line; the velocity of the body, and the resistance of the medium in each place. Plate 2. Figure 3.

Let PQ be a plane perpendicular to the plane of the scheme itself; PFHQ a curve line meeting that plane in the points P and Q; G, H, I, K four places of the body going on in this curve from F to Q; and GB, HC, ID, KE four parallel ordinates let fall from these points to the horizon, and standing on the horizontal line PQ at the points B, C, D, E; and let the distances BC, CD, DE, of the ordinates be equal among themselves. From the points G and H let the right lines GL, HN, be drawn touching the curve in G and H, and meeting the ordinates CH, DI, produced upwards, in L and N; and compleat the parallelogram HCDM. And the times, in which the body describes the arcs GH, HI, will be in a subduplicate ratio of the altitudes LH, NI, which the bodies would describe in those times, by falling from the tangents; and the velocities will be as the lengths described GH, HI directly and the times inversely. Let the times be expounded by T and t, and the velocities by GH/T and HI/t, and the decrement of the velocity produced in the time t will be expounded by GH/T - HI/t. This decrement arises from the resistance which retards the body, and from the gravity which accelerates it. Gravity, in a falling body, which in its fall describes the space NI, produces a velocity, with which it would be able to describe twice that space in the same time, as Galileo has demonstrated; that is, the velocity 2NI/t : but if the body describes the arc HI, it augments that arc only by the length HI - HN, or MI * NI / HI; and therefore generates only the velocity 2.MI * NI / (t * HI). Let this velocity be added to the beforementioned decrement, and we shall have the decrement os the velocity arising from the resistance alone, that is, GH / T - HI / t + 2.MI * NI / (t * HI). Therefore since in the same time, the action of gravity generates, in a falling body, the velocity 2.NI / t ; the resistance will be to the gravity as GH / T - HI / t + 2.MI * NI / (t * HI) to 2.NI / t, or as t * GH / T - HI + 2.MI * NI / HI to 2.NI.

Now for the abscissa's CB, CD, CE put -o, o, 2.o. For the ordinate CH put P; and for MI put any series Q.o + R.o² + S.o³ + &c. And all the terms of the series after the first, that is, R.o² + S.o³ + &c. will be NI; and the ordinates DI, EK and BG will be P - Q.o - R.o² - S.o³ - &c. P - 2.Q.o - 4.R.o² - 8.S.o³ - &c. and P + Q.o - Ro² - S.o³ - &c. respectively. And by squaring the differences of the ordinates BG - CH and CH - DI, and to the squares thence produced adding the squares of BC and CD themselves, you will have o.o + Q.Q.o.o - 2.Q.R.o³ + &c. and o.o + Q.Q.o.o + 2.Q.R.o³ + &c. the squares of the arcs GH, HI; whose roots o.√1 + Q.Q - Q.R.o.o / √1 + Q.Q, and o.√1 + Q.Q + Q.R.o.o / √1 + Q.Q are the arcs GH and HI. Moreover, if from the ordinate CH there be subducted half the sum of the ordinates BG and DI, and from the ordinate DI there be subducted half the sum of the ordinates CH and EK, there will remain R.o.o and R.o.o + 3.S.o³ the versed sines of the arcs GI and HK. And these are proportional to the lineolæ LH and NI, and therefore in the duplicate ratio of the infinitely small times T and t: and thence the ratio t / T is √(R + 3.S.o) / R or (R + frac12;S.o) / R; and t * GH / T - HI + 2.MI * NI / HI, by substituting the values of t / T, GH, HI, MI and NI just found, becomes (3.S.o.o / 2R).√1 + Q.Q to 2.R.o.o. And since 2.NI is 2.R.o.o, the resistance will be now to the gravity as (3.S.o.o / 2R).√1 + Q.Q to 2.R.o.o, that is, as 3.S.√1 + Q.Q to 4.R.R.

And the velocity will be such, that a body going off therewith from any place H, in the direction of the tangent HN, would describe, in vacuo, a Parabola, whose diameter is HC, and its latus rectum HN² / NI or (1 + Q.Q) / R.

And the resistance is as the density of the medium and the square of the velocity conjunctly; and therefore the density of the medium is as the resistance directly, and the square os the velocity inversely; that is, as 3.S.√1 + Q.Q / 4.R.R. and (1 + Q.Q) / R inversely; that is, as S / (R.√1 + Q.Q). Q.E.I.

Corollary 1. If the tangent HN be produced both ways, so as to meet any ordinate AF in T: HT / AC will be equal to √1 + Q.Q, and therefore in what has gone before may be put for √1 + Q.Q. By this means the resistance will be to the gravity as 3.S * HT to 4R.R * AC; the velocity will be as HT / (AC.√R), and the density of the medium will be as S * AC / (R * HT).

Corollary 2. And hence, if the curve line PFHQ be defined by the relation between the base or abscissa AC and the ordinate CH, as is usual; and the value of the ordinate be resolved into a converging series: The problem will be expeditiously solved by the first terms of the series; as in the following examples.

Example 1. Let the line PFHQ be a semicircle described upon the diameter PO; to find the density of the medium that shall make a projectile move in that line.

Bisect the diameter PQ in A; and call AQ, n; AC, a; CH, e; and CD, o: then DI² or AQ² - AD² = nn - aa - 2.ao - oo, or ee - 2.ao - oo; and the root being extracted by our method, will give DI = e - ao/e - oo/2e - aaoo/2e³ - ao³/2e³ - a³.o³/2.e5 - &c. Here put nn for ee + aa, and DI will become e - ao/e - nnoo/2e³ - anno³/2e5 - &c.

Such series I distinguish into successive terms after this manner: I call that the first term, in which the infinitely small quantity o is not found; the second, in which that quantity is of one dimension only; the third, in which it arises to two dimensions; the fourth, in which it is of three; and so ad infimitum. And the first term, which here is e, will always denote the length of the ordinate CH, standing at the beginning of the indefinite quantity o. The second term, which here is ao/e, will denote the difference between CH and DN; that is, the lineola MN which is cut off by compleating the parallelogram HCDM; and therefore always determines the position of the tangent HN; as, in this case, by taking MN to HM as ao/e to o, or a to e. The third term, which here is nnoo/2e³, will represent the lineola IN, which lies between the tangent and the curve; and therefore determines the angle of contact IHN, or the curvature which the curve line has in H. If that lineola IN is of a finite magnitude, it will be express'd by the third term together with those that follow in infinitum. But if that lineola be diminished in infinitum, the terms following become infinitely less than the third term, and therefore may be neglected. The fourth term determines the variation of the curvature; the fifth, the variation of the variation; and so on. Whence, by the way, appears no contemptible use of these series in the solution of problems that depend upon tangents, and the curvature of curves.

Now compare the series e - ao/e - nnoo/2e³ - anno³/2e5 - &c. with the series P - Qo - Roo - So³ - &c. and for P, Q, R and S put e, a/e, nn/2e³ and ann/2e5, and for √1 + Q.Q put √1 + aa/ee or n/e; and the density of the medium will come out as a/ne, that is, (because n is given) as a/e, or AC / CH, that is, as that length of the tangent HT, which is terminated at the semidiameter AF standing perpendicularly on PQ: and the resistance will be to the gravity as 3.a to 2.n, that is, as 3.AC to the diameter PQ of the circle; and the velocity will be as √CH. Therefore if the body goes from the place F, with a due velocity, in the direction of a line parallel to PQ, and the density of the medium in each of the places H is as the length of the tangent HT, and the resistance also in any place H is to the force of gravity as 3.AC to PQ, that body will describe the quadrant FHQ of a circle. Q.E.I.

But if the same body should go from the place P, in the direction of a line perpendicular to PO, and should begin to move in an arc of the semi-circle PFQ, we must take AC or a on the contrary side of the centre A; and therefore its sign must be changed, and we must put -a for +a. Then the density of the medium would come out as -a/e. But nature does not admit of a negative density, that is, a density which accelerates the motion of bodies; and therefore it cannot naturally come to pass, that a body by ascending from P mould describe the quadrant PF of a circle. To produce such an effect, a body ought to be accelerated by aa impelling medium, and not impeded by a resisting one.

Example 2. Let the line PFQ be a Parabola, having its axis AF perpendicular to the horizon PQ; to find the density of the medium, which will make a projectile move in that line. Figure 4.

From the nature of the Parabola, the rectangle PDQ is equal to the rectangle under the ordinate DI and some given right line: that is, if that right line be called b; PC, a; PQ, c; CH, e; and CD, o; the rectangle a + o into c - a - o or ac - aa - 2ao + co - oo is equal to the rectangle b into DI, and therefore DI is equal to (ac - aa)/b + (c - 2a)o/b - oo/b. Now the second term (c - 2a)o/b of this series is to be put for Qo, and the third term oo/b for Roo. But since there are no more terms, the coefficient S of the fourth term will vanish; and therefore the quantity S / (R.√1 + QQ), to which the density of the medium is proportional, will be nothing. Therefore, where the medium is of no density, the projectile will move in a Parabola; as Galileo hath heretofore demonstrated. Q.E.I.

Example 3. Let the line AGK be an Hyperbola, having its asymptote NX perpendicular to the horizontal plane AK; to find the density of the medium, that will make a projectile move in that line. Figure 5.

Let MX be the other asymptote, meeting the ordinate DG produced in V; and from the nature of the Hyperbola, the rectangle of XV into VG will be given. There is also given the ratio of DN to VX, and therefore the rectangle of DN into VG is given. Let that be bb: and, compleating the parallelogram DNXZ, let BN be called a; BD, o; NX, c; and let the given ratio of VZ to ZX or DN be m/n. Then DN will be equal to a - o, VG equal to bb / (a - o), VZ equal to (m/n) * (a - o), and GD or NX - VZ - VG equal to c - (m/n).a + (m/n).o - bb/(a - o). Let the term bb / (a-o) be resolved into the converging series bb/a + (bb/aa).o + (bb/a³).oo + (bb/a4).o³ + &c. and GD will become equal to c - (m/n).a - bb/a + (m/n).o - (bb/a³).o³ - (bb/4).o³ + &c. The second term (m/n).o - (bb/aa).o of this series is to be used for Q.o, the third (bb/a³).o² with its sign changed for R.o², and the fourth (bb/a4).o³ with its sign changed also for S.o³, and their coefficients m/n - bb/aa, bb/a³ and bb/a4 are to be put for Q, R and S in the former rule. Which being done, the density of the medium will come out as or, that is, if in VZ you take VT equal to VG, as 1 / XT. For aa and (m²/n²).a² - 2mbb / n + b4 / aa are the squares of XZ and ZT. But the ratio of the resistance to gravity is found to be that of 3.XT to 2.TG; and the velocity is that with which the body would describe a Parabola, whose vertex is G, diameter DG, latus rectum XT² / VG. Suppose therefore that the densities of the medium in each of the places G are reciprocally as the distances XT, and that the resistance in any place G is to the gravity as 3.XT to 2.TG; and a body let go from the place A, with a due velocity, will describe that Hyperbola AGK. Q.E.I.

Example 4. Suppose indefinitely, the line AGK to be an Hyperbola, described with the centre X, and the asymptotes MX, NX, so that, having constructed the rectangle XZDN, whose side ZD cuts the Hyperbola in G and its asymptote in V, VG may be reciprocally as any power DNn of the line ZX or DN, whose index is the number n: To find the density of the medium in which a projected body will describe this curve. Figure 5.

For BN, BD, NX put A, O, C respectively, and let VZ be to XZ or DN as d to e, and VG be equal tobb / DNn; then DN will be equal to A - O, VG = bb / (A - O)n, VZ = (d/e).A - O, and GD or NX - VZ - VG equal to C - (d/e).A + (d/e).O - bb / A - On. Let the term bb / (A - O)n be resolved into an infinite series

and GD will be equal to

The second term of this series is to be used for Q.o, the third  for R.oo, the fourth  for S.o³. And thence the density of the medium S / (R. √1 + QQ), in any place G, will be, and therefore if in VZ you take VT equal to n * VG, that density is reciprocally as XT. For A² and  are the squares of XZ and ZT. But the resistance in the same place G is to the force of gravity as 3S * XT / A to 4RR, that is, as XT to. And the velocity there, is the same wherewith the projected body would move in a Parabola, whose vertex is G, diameter GD, and latus rectum (1 + QQ) / R or. Q.E.I.

In the same manner that the density of the medium comes out to be as (S x AC) / (R x HT), in corollay 1. if the resistance is put as any power Vn of the velocity V, the density of the medium will come out to be as Figure 3.

And therefore if a curve can be found, such that the ratio of, or of S²/R4-n to (1 + QQ)n-1 may be given: the body, in an uniform medium, whose resistance is as the power Vn of the velocity V, will move in this curve. But let us return to more simple curves.

Because there can be no motion in a Parabola except in a non-resisting medium, but in the Hyperbola's here described 'tis produced by a perpetual resistance; it is evident that the line which a projectile describes in an uniformly resisting medium, approaches nearer to these Hyperbola's than to a Parabola. That line is certainly of the hyperbolic kind, but about the vertex it is more distant from the asymptotes, and in the parts remote from the vertex draws nearer to them, than these Hyperbola's here described. The difference however is not so great between the one and the other, but that these latter may be commodiously enough used in practice instead of the former. And perhaps these may prove more useful, than an Hyperbola that is more accurate, and at the same time more compounded. They may be made use of then in this manner. Figure 5.

Compleat the parallelogram XYGT, and the right line GT will touch the hyperbola in G, and therefore the density of the medium in G is reciprocally as the tangent GT, and the velocity there, as √GT² / GV² and the resistance is to the force of gravity as GT to {(2nn + 2n) / (n + 2)} * GV.

Therefore if a body projected from the place A in the direction of the right line AH, (Figure 6.) describes the Hyperbola AGK, and AH produced meets the asymptote NX in H, and AI drawn parallel to it meets the other asymptote MX in I; the density of the medium in A will be reciprocally as AH, and the velocity of the body as √AH² / AI, and the resistance there to the force of gravity as AH to {(2nn + 2n) / (n + 2)} * AI. Hence the following rules are deduced.

Rule 1. If the density of the medium at A, and the velocity with which the body is projected remain the same, and the angle NAH be changed; the lengths AH, AI, HX will remain. Therefore if those lengths, in any one case, are found, the Hyperbola may afterwards be easily determined from any given angle NAH.

Rule 2. If the angle NAH and the density of the medium at A remain the same, and the velocity with which the body is projected be changed, the length AH will continue the same; and AI will be changed in a duplicate ratio of the velocity reciprocally.

Rule 3. If the angle NAH, the velocity of the body at and the accelerative gravity remain the same, and the proportion of the resistance at A to the motive gravity be augmented in any ratio; the proportion of AH to AI will be augmented in the same ratio, the latus rectum of the abovementioned Parabola remaining the same, and also the length AH²/AI proportional to it; and therefore AH will be diminished in the same ratio, and AI will be diminished in the duplicate of that ratio. But the proportion of the resistance to the weight is augmented, when either the specific gravity is made less, the magnitude remaining equal, or when the density of the medium is made greater, or when, by diminishing the magnitude, the resistance becomes diminished in a less ratio than the weight.

Rule 4. Because the density os the medium is greater near the vertex of the Hyperbola, than it is in the place A; that a mean density may be preserv'd, the ratio of the least of the tangents GT to the tangent AH ought to be found, and the density in A augmented in a ratio a little greater than that of half the sum of those tangents to the least of the tangents GT.

Rule 5. If the lengths AH, AI are given, and the figure AGK is to be described: produce HN to X, so that HX may be to AI as n+1 to 1; and with the centre X, and the asymptotes MX, NX describe an Hyperbola thro' the point A, such that AI may be to any of the lines VG as XVn to XIn.

Rule 6. By how much the greater the number n is, so much the more accurate are these Hyperbola's in the ascent of the body from A, and less accurate in its descent to K; and the contrary. The Conic Hyperhola keeps a mean ratio between these, and is more simple than the rest. Therefore if the Hyperbola be of this kind, and you are to find the point K, where the projected body falls upon any right line AN passing thro' the point A: let AN produced meet the asymptotes MX, NX in M and N, and take NK equal to AM.

Rule 7. And hence appears an expeditious method of determining this Hyperbola from the phenomena. Let two similar and equal bodies be projected with the same velocity, in different angles HAK, hAk, (Figure 6.) and let them fall upon the plane of the horizon in K and k; and note the proportion of AK to Ak- Let it be as d to e. Then erecting a perpendicular AI of any length, assume any how the length AH or Ah, and thence graphically, or by scale and compass, collect the lengths AK, Ak (by rule 6.) If the ratio of AK to Ak be the same with that of d to e, the length of AH was rightly assumed. If not, take on the indefinite right line SM, (Figure 7.) the length SM equal to the assumed AH; and erect a perpendicular MN, equal to the difference AK/Ak - d/e of the ratio's drawn into any given right line. By the like method, from several assumed lengths AH, you may find several points N; and draw thro' them all a regular curve NNXN, cutting the right line SMMM in X. Lastly, assume AH equal to the abscissa SX, and thence find again the length AK; and the lengths, which are to the assumed length AI and this last AH, as the length AK known by experiment, to the length AK last found, will be the true lengths AI and AH, which were to be found. But these being given, there will be given also the resisting force of the medium in the place A, it being to the force of gravity as AH to 2AI. Let the density of the medium be increased by rule 4. and if the resisting force just found be increased in the same ratio, it will become still more accurate.

Rule 8. The lengths AH, HX being found; let there be now required the position of the line AH, according to which a projectile thrown with that given velocity, shall fall upon any point K. At the points A and K, (Figure 6.) erect the lines AC, KF perpendicular to the horizon; whereof let AC be drawn downwards, and be equal to AI or ½HX. With the asymptotes AK, KF, describe an Hyperbola, whose conjugate shall pass thro* the point C; and from the centre A, with the interval AH, describe a circle cutting that Hyperbola in the point H; then the projectile thrown in the direction of the right line AH will fall upon the point K. Q.E.I. For the point H, because of the given length AH, must be somewhere in the circumference of the described circle. Draw CH meeting AK and KF in E and F; and because CH, MX are parallel, and AC, AI equal, AE will be equal to AM, and therefore also equal to KN. But CE is to AE as FH to KN, and therefore CE and FH are equal. Therefore the point H falls upon the hyperbolic curve described with the asymptotes AK, KF, whose conjugate passes thro' the point C; and is therefore found in the common intersection of this hyperbolic curve and the circumference of the described circle. Q.E.D. It is to be observed that this operation is the same, whether the right line AKN be parallel to the horizon, or inclined thereto in any angle; and that from two intersections H, H, there arise two angles NAH, NAH; and that in mechanical practice it is sufficient once to describe a circle, then to apply a ruler CH, of an indeterminate length, so to the point C, that its part FH, intercepted between the circle and the right line FK, may be equal to its part CE placed between the point C and the right line AK.

What has been said of Hyperbola's may be easily applied to Parabola's. For if (Figure 8.) a Parabola be represented by XAGK, touched by a right line XV in the vertex X; and the ordinates IA, VG be as any powers XIn, XVn of the abscissa's XI, XV; draw XT, GT, AH, whereof let XT be parallel to VG, and let GT, AH touch the Parabola in G and A: and a body projected from any place A, in the direction of the right line AH, with a due velocity, will describe this Parabola, if the density of the medium in each of the places G, be reciprocally as the tangent GT. In that case the velocity in G will be the same as would cause a body, moving in a non-resisting space, to describe a Conic Parabola, having G for its vertex, VG produced downwards for its diameter, and 2GT² / (nn - n * VG) for its latus rectum. And the resisting force in G will be to the force of gravity, as GT to {(2nn - 2n) / (n - 2)} * VG. Therefore if NAK represent an horizontal line, and, both the density of the medium at A and the velocity with which the body is projected, remaining the same, the angle NAH be any how alter'd; the lengths AH, AI, HX will remain; and thence will be given the vertex X of the Parabola, and the position of the right line XI, and by taking VG to IA as XVn to XIn, there will be given all the points G of the Parabola, thro* which the projectile will pass.


text:
The Inner Life, v. I/Fifth Section/V

THE DESIRE-ELEMENTAL Much of the matter of the astral body is vivified by elemental essence, which is cut off for the time being from the general mass which belongs to the plane, and becomes the man's expression on that plane. This is a living, though not an intelligent essence. But it has a kind of instinct which Mr. Sinnett calls “dawning intelligence,” which guides it into getting what it wants. Blindly and without reason, but instinctively, it seeks its ends, and shows great ingenuity in obtaining its desires and in furthering its evolution. Evolution for it is a descent into matter; its aim is to become a mineral monad. Therefore, its object in life is to get as near to the physical plane as it can, to come into contact with as many of the vibrations of the coarser kind as possible. It knows nothing of you; it could not know or imagine anything of you; but it does realize that it is apart from the general stock, and that it is good to be apart. It is not a devil, and you must not get the idea that it is to be hated. It is part of the Divine Life, just as you are; but its interests are diametrically opposed to yours. It wants to evolve downwards; you want to evolve upwards. It desires to preserve its separate life, and it feels that it can do so only by means of its connection with you. It is conscious of a something which is your lower mind, and realizes that if it can englobe, as it were, this mind, and persuade you that its and yours interests are one, you will increasingly supply it with the sensations it desires. When it gets the matter sufficiently entangled to suit its purpose, you cannot withdraw it, the result being that some of this matter of the lower mind is then lost to you altogether in the life after death. So, you see, here is the desire-elemental seeking its own ends; not knowing that it is injuring you by trying to entangle your lower mind. The more it can do this the better for it, for the more mental matter it can entangle the longer will be its astral life — that life still enduring even after you have passed into the heaven-world. In Theosophical phraseology it has been known as the shade. Your business is not to allow yourself to be deceived; it understands nothing of your evolution, and is not responsible for it; it simply tries to turn you to its own purpose. You ought to understand the situation, and refuse to be drawn. Do let us realize this: that this elemental is not ourselves. It is never you who desire these lower things, but this creature. It is not so much that we have to make a great fight against it, but we should shake ourselves free, saying: “This is not I; I do not want this lower thing.” Somebody wants it. Yes, it is this elemental; and you are responsible for its likes and wants, for in your last life you made it what it is. Not that this particular collection of astral matter and elemental essence existed then; it did not, for it was newly gathered together at your birth this time. But it is an exact reproduction of the matter in your astral body at the end of your last astral life. Nevertheless it is not you; and you must ever bear this in mind all through life, and even more during the life after death, for then it has still greater power to deceive you. But you may think that by thus refusing to allow it to influence you, you are checking its evolution. Not at all. You are doing better for the elemental if you control the lower passions, and take a firm stand of your own. It is true you do not develope a very low part of it; but you may drop the lower and evolve the higher. An animal can supply the lower kinds of vibrations even better than you can yourself, whereas none but man can evolve the higher type of essence. After the death of the physical body the ordinary man, who has never heard all this, finds himself when he wakes up on the other side in a totally unexpected condition of affairs, and is generally more or less disturbed thereby. Finally, he accepts these conditions which he does not understand, thinking them necessary and inevitable. Some no doubt are, but some are not, and with knowledge the latter could be transcended. The elemental is afraid, because it knows that the death of the physical body means that the term of its separated life is limited; it knows that the man's astral death will more or less quickly follow, and with it the loss to it of vivid and intense sensations. Consequently it adopts the best plan it can think of for the preservation of the man's astral body. It evidently knows enough of astral physics to realize that the coarsest matter can hold together longest, and best stand friction. So it arranges the matter in rings, the coarsest on the outside. And in so doing it is right, from its point of view. During physical life the astral body is like swirling, boiling water, but after death it arranges the matter in a series of graduated sheaths, so that full circulation is impossible. Now there are no sense organs in the astral body. There are in it organs corresponding to the physical sense-organs, but you do not see, hear and smell with them. You hear and see all over the surface of the body. Each sub-plane has its own matter; and it is by means of the matter of that sub-plane in your body that you can respond to its vibrations. Whatever matter is on the outside (or surface) of your body responds to these vibrations, and you see or hear by it alone. Consequently, what has happened is this: the elemental has, by this arrangement of the matter of your body, shut you up, as it were, in a box of astral matter, which enables you to see and hear things of the lowest and coarsest plane only. If you object to being shut up in this way, it endeavours to make you believe that unless you do thus firmly root yourself into the lower matter you will float off, and lose yourself in a nebulous vagueness. But if, on the other hand, you were to set your will to oppose it, then at once there would be a difference. The particles of the astral body would be kept all intermingled, as in life; and you would, in consequence, be free of all the sub-planes. The final struggle with it takes place at the conclusion of the astral life, for then the ego endeavours to draw back into himself all that he put down into incarnation at the beginning of the life which has just closed — to recover as it were the principal which he has invested, plus the interest of the experience which has been gained and the qualities which have been developed during that life. But when he attempts to do this he is met with determined opposition from this desire-elemental, which he himself has created and fed. Though it can hardly be described as intelligent, it has a strong instinct of self-preservation, which leads it to resist with all the force at its command the extinction which threatens it. In the case of all ordinary mortals it attains a certain measure of success in its efforts, for much of the mental faculty has during life been governed by the lower desires and prostituted to their service, or in other words the lower mind has been so seriously entangled by desire that it is impossible for it to be entirely freed. The result of the struggle is therefore that some portion of the mental matter and even of causal matter is retained in the astral body after the ego has completely broken away from it. When a man has during life completely conquered his lower desires and succeeded in absolutely freeing the lower mind from desire, there is practically no struggle, and the ego reclaims in full both principal and interest; but there is unfortunately an opposite extreme when he is able to reclaim neither. So our business, both during life and after death, is to control this desire-elemental, and not let it control us. Realize that you are a god in the making. All the power and force of the universe are on your side. The result is certain. Range yourself on the side of the Law, and all will be simplified. Absolute control of passions is eminently desirable, but is obtained by few. You have to keep your temper on the astral plane. You see many dreadful things, and if you have not all feelings thoroughly under control you may easily do something for which you will be sorry. Down here people often commit casual brutality and think nothing of it; a callous schoolmaster, for example, beats a child without realizing his wickedness; but on the astral plane the heinousness of such a crime is at once obvious, and even the awful horrors of the karma which it entails may often be seen. On the astral you see the full effects of even an unkind word. Tremendous and violent passions may often attract low kinds of beings, who enter into the thought-forms and enjoy the vibrations. Such animated thought-forms may last for years, and even produce poltergeist phenomena.

title: The Inner Life, v. I/Fifth Section/V