which it is inscribed so that the Complement of the Spiral that is that space in the Circle which is without the Spiral Line is double to the space within the Spiral Line In the same manner if there be taken a mean proportional every where between the Semidiameter of the Circle which contains the Spiral and that part of the Semidiameter which is within the same there will be made another figure which will be half the Circle And to conclude this Rule serves for all such Spaces as may be described by a Line or Superficies decreasing either in magnitude or power so that if the proportions in which they decrease be commensurable to the proportions of the times in which they decrease the magnitudes of the figures they describe will be known 12 The truth of that proposition which I demonstrated in the second Article which is the foundation of all that has been said concerning Deficient Figures may be derived from the Elements of Philosophy as having iâ—â— original in this That all equality and inequality between two effects that is all Proportion proceeds from and is determined by the equal and unequal causes of those effects or from the proportion which the causes concurring to one effect have to the causes which concurre to the producing of the other effect and that therefore the proportions of Quantities are the same with the proportions of their causes Seeing therefore two Deficient Figures of which one is the Complement of the other are made one by motion decreasing in a certain time and proportion the other by the loss of Motion in the same time the causes which make and determine the quantities of both the figures so that they can be no other then they are differ onely in this that the proportions by which the quantity which generates the figure proceeds in describing of the same that is the proportions of the remainders of all the times and altitudes may be other proportions then those by which the same generating quantity decreases in making the Complement of that Figure that is the proportions of the quantity which generates the Figure continually diminished Wherefore as the proportions of the quantity in which Motion is lost is to that of the decreasing quantities by which the Deficient Figure is generated so will the Defect or Complement be to the Figure it self which is generated 13 There are also other quantities which are determinable from the knowledge of their causes namely from the comparison of the Motions by which they are made and that more easily then from the common Elements of Geometry For example That the Superficies of any portion of a Sphere is equal to that Circle whose Radius is a straight Line drawn from the Pole of the portion to the Circumference of its base I may demonstrate in this manner Let B A C in the 7 Figure be a portion of a Sphere whose Axis is A E whose base is B C let A B be the straight line drawn from the Pole A to the base in B and let A D equal to A B touch the great Circle B A C in the Pole A. It is to be proved that the Circle made by the Radius A D is equal to the Superficies of the portion B A C. Let the plain A E B D be understood to make a revolution about the Axis A E it is manifest that by the straight line A D a Circle will be described and by the arch A B the Superficies of a portion of a Sphere and lastly by the Subtense A B the Superficies of a right Cone Now seeing both the straight line A B and the arch A B make one and the same revolution and both of them have the same extreme points A and B the cause why the the Spherical Superficies which is made by the arch is greater then the Conical Superficies which is made by the Subtense is that A B the arch is greater then A B the Subtense and the cause why it is greater consists in this that although they be both drawn from A to B yet the Subtense is drawn straight but the arch angularly namely according to that angle which the arch makes with the Subtense which angle is equal to the angle D A B for an angle of contingence adds nothing to an angle of a Segment as has been shewn in the 14 Chapter at the 16th Article Wherefore the magnitude of the angle D A B is the cause why the Superficies of the portion described by the arch A B is greater then the Superficies of the right Cone described by the Subtense A B. Again the cause why the Circle described by the Tangent A D is greater then the Superficies of the right Cone described by the Subtense A B notwitstanding that the Tangent and the Subtense are equal and both moved round in the same time is this that A D stands at right angles to the Axis but A B obliquely which obliquity consists in the same angle D A B. Seeing therefore the quantity of the angle D A B is that which makes the excess both of the Superficies of the Portion and of the Circle made by the Radius A D above the superficies of the Right Cone described by the subtense A B it follows that both the Superficies of the Portion and that of the Circle do equally exceed the Superficies of the Cone Wherefore the Circle made by A D or A B and the Spherical Superficies made by the arch A B are equal to one another which was to be proved â—4 If these Deficient Figures which I have described in a 〈◊〉 were capable of exact description then any number of mean proportionals might be found out between two straight lines given For example in the Parallelogram A B C D in the 8th Figure let the three-sided figure of two Means be described which many call a Cubical Parabola and let R and S be two given straight lines between which if it be required to find two mean proportionals it may be done thus Let it be as R to S so B C to B F and let F E be drawn parallel to B A and cut the crooked line in E then through E let G H be drawn parallel and equal to the straight line A D and cut the Diagonal B D in I for thus we have G I the greatest of two Means between G H and G E as appears by the description of the figure in the 4th Article Wherefore if it be as G H to G I so R to another line T that T will be the greatest of two Means between R and S. And therefore if it be again as R to T so T to another line X that will be done which was required In the same manner four mean proportionals may be found out by the description of a three-sided figure of four Means and so any other number of Means c. CHAP. XVIII Of the Equation of Straight Lines with the
to it self so that in what part soever of the Ecliptick the Center of the Epicycle be found and in what part soever of the Epicycle the Center of the Earth be found at the same time the Axis of the Earth will be parallel to the place where the same Axis would have been if the Center of the Earth had never gone out of the Ecliptick Now as I have demonstrated the simple annual motion of the Earth from the supposition of Simple Motion in the Sunne so from the supposition of Simple Motion in the Earth may be demonstrated the monethly Simple Motion of the Moon For if the names be but changed the Demonstration will be the same and therefore need not be repeated 7 That which makes this supposition of the Sunnes Simple Motion in the Epicycle fghi probable is First that the Periods of all the Planets are not onely described about the Sunne but so described as that they are al contained within the Zodiack that is to say within the latitude of about 16 degrees for the cause of this seems to depend upon some power in the Sunne especially in that part of the Sunne which respects the Zodiack Secondly that in the whole coâ—passe of the heavens there appears no other Body from which the cause of this Phaenomenon can in probability be derived Besides I could not imagine that so many and such various motions of the Planets should have no dependance at all upon one another But by supposing motive power in the Sunne we suppose motion also for power to move without motion is no power at all I have therefore supposed that there is in the Sunne for the governing of the primary Planets and in the Earth for the governing of the Moon such motion as being received by the primary Planets and by the Moon makes them necessarily appear to us in such manner as we see them Whereas that circular motion which is commonly attributed to them about a fixed Axis which is called Conversion being a motion of their parts onely and not of their whole Bodies is insufficient to salve their Appearances For seeing whatsoever is so moved hath no endeavour at all towards those parts which are without the circle they have no power to propagate any endeavour to such Bodies as are placed without it And as for them that suppose this may be done by Magnetical Virtue or by incorporeall and immateriall Species they suppose no naturall cause nay no cause at all For there is no such thing as an Incorporeal Movent and Magnetical Virtue is a thing altogether unknown and whensoever it shall be known it will be found to be a motion of Body It remaines therefore that if the primary Planets be carried about by the Sunne and the Moon by the Earth they have the simple circular motions of the Sunne and the Earth for the causes of their circulations Otherwise if they be not carried about by the Sunne and the Earth but that every Planet hath been moved as it is now moved ever since it was made there will be of their motions no cause naturall For either these motions were concreated with their Bodies and their cause is supernatural or they are coeternal with them and so they have no cause at all For whatsoever is Eternall was never generated I may add besides to confirme the probability of this simple motion that allmost all learned men are now of the same opinion with Copernicus concerning the parallelisme of the Axis of the Earth it seemed to me to be more agreeable to truth or at least more handsome that it should be caused by simple Circular Motion alone than by two motions one in the Ecliptick and the other about the Earths own Axis the contrary way neither of them Simple nor either of them such as might be produced by any motion of the Sunne I thought best therefore to retain this Hypothesis of Simple Motion and from it to derive the causes of as many of the Phaenomena as I could and to let such alone as I could not deduce frm thence It will perhaps be objected that although by this supposition the reason may be given of the Parallelisme of the Axis of the Earth and of many other Appearances nevertheless seeing it is done by placing the Body of the Sunne in the Center of that Orbe which the Earth describes with its annual motion the supposition it self is false because this annual Orbe is excentrique to the Sunne In the first place therefore let us examine what that Excentricity is and whence it proceeds 8 Let the annual Circle of the Earth abcd in the same 3d figure be divided into four equal parts by the straight lines ac bd cutting one another in the Center e and let a be the beginning of Libra b of Capricorn c of Aries and d of Cancer and let the whole Orbe abcd be understood according to Copernicus to have every way so great distance from the Zodiack of the fixed Starres that it be in comparison with it but as a point Let the Earth be now supposed to be in the beginning of Libra at a. The Sunne therefore will appear in the beginning of Aries at c. Wherefore if the Earth be moved from a to b the apparent motion of the Sunne will be from c to the beginning of Cancer in d and the Earth being moved forwards from b to c the Sunne also will appear to be moved forwards to the beginning of Libra in a Wherefore cda will be the Summer Arch and the Winter Arch will be abc Now in the time of the Suns apparent motion in the Summer Arch there are numbred 186¾ dayes and consequently the Earth makes in the same time the same number of diurnal conversions in the Arch abc and therefore the Earth in its motion through the Arch cda will make onely 178½ diurnal conversions Wherefore the Arch a b c ought to be greater then the Arch c d a by 8¼ dayes that is to say by almost so many degrees Let the Arch a r as also c s be each of them an Arch of two degrees and 1 16. Wherefore the Arch r b s will be greater then the Semicircle a b c by 4 degrees and â…› and greater then the Arch s d r by 8 degrees and ¼ The Equinoxes therefore will be in the points r s and therefore also when the Earth is in r the Sunne will appear in s. Wherefore the true place of the Sunne will be in t that is to say without the Center of the Earths annual motion by the quantity of the Sine of the Arch a r or the Sine of two degrees and 16 minutes Now this Sine putting 100000 for the Radius will be neer 3580 parts thereof And so munh is the Excentricity of the Earths annual motion provided that that motion be in a perfect circle and s r are the Equinoctial points and the straight lines s r c a produced both wayes till they
of the parts of those plants made an Odorous liquour so also of aire passing through the same plants whilest they are growing are made Odorous aires And thus also it is with the Juices and Spirits which are bred in Living Creatures 16 That Odorous Bodies may be made more Odorous by Contrition proceeds from this that being broken into many parts which are all Odorous the aire which by respiration is drawn from the Object towards the Organ doth in its passage touch upon all those parts and receives their motion Now the aire toucheth the superficies onely and a Body having less superficies whilest it is whole then all its parts together have after it is reduced to powder it follows that the same Odorous Body yeildeth less Smell whilest it is whole then it will do after it is broken into smaller parts And thus much of Smels 17 The Tast follows whose generation hath this difference from that of the Sight Hearing and Smelling that by these we have Sense of remote Objects whereas we Tast nothing but what is contiguous and doth immediately touch either the Tongue or Palate or both From whence it is evident that the cuticles of the Tongue and Palate and the Nerves inserted into them are the first Organ of Tast and because from the concussion of the parts of these there followeth necessarily a concussion of the Pia Mater that the action communicated to these is propagated to the Brain and from thence to the farthest Organ namely the Heart in whose reaction consisteth the nature of Sense Now that Savours as well as Odours doe not onely move the Brain but the Stomack also as is manifest by the loathings that are caused by them both they that consider the Organ of both these Senses will not wonder at all seeing the Tongue the Palate the Nostrils have one and the same continued cuticle derived from the Dura Mater And that Effluviums have nothing to doe in the Sense of Tasting is manifest from this that there is no Tast where the Organ and the Object are not contiguous By what variety of motions the different kinds of Tasts which are innumerable may be distinguished I know not I might with others derive them from the divers figures of those Atomes of which whatsoever may be Tasted consisteth or from the diverse motions which I might by way of Supposition attribute to those Atomes conjecturing not without some likelyhood of truth that such things as tast Sweet have their particles moved with slow circular motion and their figures Spherical which makes them smooth and pleasing to the Organ that Bitter things have circular motion but vehement and their figures full of Angles by which they trouble the Organ and that Sowre things have straight and reciprocal motion and their figures long and small so that they cut and wound the Organ And in like manner I might assigne for the causes of other Tasts such several motions and figures of Atomes as might in probability seem to be the true causes But this would be to revolt from Philosophy to Divination 18 By the Touch we feel what Bodies are Cold or Hot though they be distant from us Others as Hard Soft Rough and Smooth we cannot feel unless they be contiguous The Organ of Touch is every one of those membranes which being continued from the Pia Mater are so diffused throughout the whole Body as that no part of it can be pressed but the Pia Mater is pressed together with it Whatsoever therefore presseth it is felt as Hard or Soft that is to say as more or less Hard. And as for the Sense of Rough it is nothing else but innumerable perceptions of Hard and Hard succeeding one another by short intervals both of time and place For we take notice of Rough and Smooth as also of Magnitude and Figure not onely by the Touch but also by Memory For though some things are touched in one Point yet Rough and Smooth like Quantity and Figure are not perceived but by the Flux of a Point that is to say we have no Sense of them without Time and we can have no Sense of Time without Memory CHAP. XXX Of Gravity 1 A Thick Body doth not contain more Matter unless also more Place then a Thinne 2 That the Descent of Heavy Bodies proceeds not from their own Appetite but from some Power of the Earth 3 The difference of Gravities proceedeth from the difference of the Impetus with which the Elements whereof Heavy Bodies are made do fall vpon the Earth 4 The cause of the Descent of Heavy Bodies 5 In what proportion the Descent of Heavy Bodies is accelerated 6 Why those that Dive do not when they are under Water feel the waight of the Water above them 7 The Waight of a Body that floateth is equal to the Waight of so much Water as would fill the space which the immersed part of the Body takes up within the Water 8 If a Body be Lighter then Water then how big soever that Body be it will float upon any quantity of Water how little soever 9 How Water may be lifted up and forced out of a Vessel by Air. 10 Why a Bladder is Heavier when blown full of aire then when it is empty 11 The cause of the ejection upwards of Heavy Bodies from a Wind-Gun 12 The cause of the ascent of Water in a Weather-glass 13 The cause of motion upwards in Living Creatures 14 That there is in Nature a kind of Body Heavier then Aire which nevertheless is not by Sense distinguishable from it 15 Of the cause of Magnetical vertue 1 IN the 21 Chapter I have defined Thick and Thinne as that place required so as that by Thick was signified a more Resisting Body and by Thinne a Body less Resisting following the custome of those that have before me discoursed of Refraction Now if we consider the true and vulgar signification of those words we shall find them to be Names Collective that is to say Names of Multitude as Thick to be that which takes up more parts of a space given Thinne that which contains fewer parts of the same magnitude in the same space or in a space equal to it Thick therefore is the same with Frequent as a Thick Troop And Thinne the same with Unfrequent as a Thinne Rank Thinne of Houses not that there is more matter in one place then in another equal place but a greater quantity of some named Body For there is not less matter or Body indefinitely taken in a Desert then there is in a City but fewer Houses or fewer Men. Nor is there in a Thick Rank a greater quantity of Body but a greater number of Souldiers then in a Thinne Wherefore the multitude paucity of the parts contained within the same space do constitute Density and Rarity whether those parts be separated by Vacuum or by Aire But the consideration of this is not of any great moment in Philosophy and therefore I let