ANSWERS Upon several heads in PHILOSOPHY

The Seventh Answer.

THough I cannot consent unto your thoughts about the Angle of Contact, I should be angry with my own rashness, should I be forward to blame your late speculations thereabout; seeing so many noble and sprightly wits, and such as have deserved the honour of Europe, siding with you. Especially I ho­nour your thorow-conversation in the learned persons works, whom I perceive you so large­ly [Page 83] to have gleaned after: you could not have plowed in these hard grounds with a more worthy Assistant. But knowing it is not Com­plement you expect in Philosophy but Reason; I shall in as short, as the matter will bear, and as plainly as I can, summe up the grounds, why I as yet remain so farre dissatisfyed, that I cannot without injustice to my own judgment, desert the generous and learned dissenters from your Opinion: or cease to averre Angles of Contact to be true quantities, equal; or un­equal one to another, though the highest mul­tiplex of the greatest citradiametral Angle of Contact, be never able to exhaust, or equa­lize the least right-lined part of the most acute right-lined Angle.

And that we may not mistake each other a­bout the State of the Controversy; and to be shorter, and more clear in the insuing dis­course; we shall first set down the definitions of certain termes, about which I suppose there will be no difference between us. As.

1^st. (Our discourse extending no further then to plain Angles) that by a Right-lined angle, we understand a plane angle contained under two concurring sides, or lines, which are two right-lines not posited directly the one in the production of the other, nor parallel the one to the other. As in fig. 1. the Angle BAC is a right-lined angle contained under the two [Page 84] right lines BA and CA; BA and CA not being so posited at A as to make one right line: nor can they be parallel because of their rectitude and concurrence.

2^ly. That by a crooked-lined Angle, we understand a plane Angle, contained under two concurring sides, or lines, which are two crooked lines; as in the fig. 2.3 4. the Angle BAC is a crooked lined angle contained under the two Crooked lines BA and CA.

3^ly. That by a crooked lined angle of con­caves, or a Concavo-concave angle, we un­derstand a plane angle contained under two concurring crooked lines; both which crooked lines obvert at the Angle first of all inwards, the concave side of their Curvature, towards the angle under then contained. So in fig. 4. the angle BAC is a crooked-lined angle of Conca­ves, or a Concavo-concave angle, contained under the two crooked lines BA and CA both which obvert the Concave side of their cur­vature towards the angle under then contained.

4^ly. That by a crooked-lined angle of Con­vexes, or a Convexo-convexe angle, we un­derstand a plane angle, contained under two concurring crooked-lines; both which croo­ked-lines obvert at the angle first of all in­wards, the Convexe sides of their Curvature towards the angle, under them contained. So in fig. 2. the angle BAC. is a crooked-lined [Page 85] angle of convexes, or a Convexo-convexe angle, contained under the two crooked-lines BA and CA, both which obvert the convexe sides of their curvature, first of all inwards, towards the Angle under them contained.

5^ly. That by a concavo-convexe angle we understand a plane angle, contained under two concurring crooked lines; the one of which, at the angle first of all, obverts the convexe side of its curvature inwards towards the con­tained angle, the other the concave side of its curvature. So in fig. the 3d. the angle BAC is a concave-convexe, or a convexo-concave crooked-lined angle, contained under the two crooked lines BA and CA, whereof the one BA obverts the convexe side of its curvature towards the contained angle, but the other CA obverts the concave side of its curvature towards the same contained angle.

6^ly. That by a mixed lined angle, we un­derstand a plane angle, contained under two concurring lines; whereof the one is a right line, the other a crooked line. So in fig. 5th. 6th. the angle BAC is a mixed-lined angle, contained under the right line CA. and the crooked line BA.

7^ly. That by a Recto-concave angle, we understand a plane angle, contained under two concurring lines; whereof the one is a right line, the other a crooked line, having the con­cave [Page 86] side of its curvature obverted towards the contained angle, so in fig. 5. The angle B.A.C. is drawn a recto-concave angle, being con­tained under the right line C.A. and the croo­ked line B. obverting the concave side of its cur­vature towards the contained angle.

8^ly. That by a recto-convexe angle, we understand a plane angle, contained under two concurring lines; whereof the one is a right line, the other a crooked line, having the con­vexe side of its curvature obverted towards the angle contained, so in fig. 6th. the angle BAC. is drawn a recto-convexe angle, being contained under the right line CA. and the crooked line BA. obverting the convexe side of its curvature towards the contained angle.

9^ly. That by an Angle of contact we under­stand a plane angle, contained under two con­curring lines; which two lines are neither one and the same line produced, nor yet in their present positure and inclination can possi­bly cut one another, how farr soever either, or both of them be produced. So in fig. 7th. and fig. 8th. the angle BAC. being an angle of contact, if in either figure the crookd-line BA, be produced beyond the point of contact A, unto D, and in fig, 7th. the crooked-line CA, in figure 8th. the right line CA. be produced beyond the point of contact A unto E; in neither the line CAE can fal into the production of the line BAD, nor in either [Page 87] shall the line CAE cut the line BAD. And

10^ly. From the figuration, positure and pro­perties of the sides of such angles of contact; that by a recto-convexe angle of contact, we under­stand a plane angle being recto-convexe, as above defined, and being also an angle of contact. So in fig. 8th. if in the recto-convexe angle BAC, the right line CA touch and cut not the convexe arch BA the angle BAC is a recto-convexe angle of contact.

11^ly. That, by a recto-concave angle of contact we understand a plane angle of con­tact, which is also recto-concave, as above defined. As in fig. the 8th. if in the recto-con­cave angle EAB the right line EA touch and cut not the concave arch BA, the angle EAB is a recto-concave angle of contact.

12. That by a concavo-convexe angle of contact we understand a plane angle of Contact, being also concavo-convexe, as above defined. So in fig. 12th. if RAH being a concavo-convexe angle, the arch RA touch and cut not the arch HA then the angle RAH is a concavo-convexe angle of Contact.

13. That by a Citra-diametral-concavo-con­vexe angle of Contact, we understand a plane concavo-convexe angle of Contact, both whose conteining Arches lie on the same side of the right line tangent, which toucheth both the arches in the same point of Contact, As in fig. [Page 88] 12th. the concavo-convexe angle of Contact HAR, having both Arches HA and RA lying on the same side of the right line tangent BA, which toucheth both the Arches in the same point A, the concavo-convexe angle of Contact HAR is citradiametral.

14. That by an ultra-diametral concavo-convexe angle of Contact, we understand a plane angle of contact being concavo-convexe, and having the one arch on the one side of the diameter passing thorow the Contact point, and beyond the right line tangent which tou­cheth the Arches in the same point; but ha­ving the other arch on the other side of that diameter, and on the other side of the right line tangent. So in fig. 12th. the two Circles FAH and ADK touching in the point A, and AK being their common diameter and AB being a right line tangent to both, the concavo-convexe angle of Contact DAH is ultra-dia­metral.

15. That by a mixed crooked lined angle, we understand a plane angle contain'd under two concurring crooked lines, of such several curvatures that it is impossible for them to be coapted from the angular point the one unto the other, or the one, or its production to be coapted to the other, or its production, at and from the angular point, by any circumduction whatsoever, so in fig. 9.10.11. the angle [Page 89] BAC, being a mixed crooked lined angle, AB as being of a different curvature from AC, can no way possibly be coapted by any circumduction to AC from the angular point A: and if BA be produced to D and CA to E the crooked-line AB, because of its different curvature can neither be coapted to the crooked line AC nor to its production AE: nor the crooked line AC to the crooked line AB, nor to its production AD, by any circumduction upon the angular point whatsoever.

16. That by the isoclitical sides of an Angle, we understand two such lines, whether right, or crooked, containing a plane angle, as accor­ding to their present site and positure, without inversion when crooked, moving by the sides nearer one to another, will by such motion at last come to be coincident with and coapted exactly the one unto the other: and that by an isoclitical angle we understand a plane angle, contained under such isoclitical sides. So every right lined angle is manifestly isocliti­cal; and its sides are isoclitical, the sides by such motion on the angular point one towards ano­ther being manifestly coincidible and coaptable So also in fig. 3^d. if in the concavo-convexe croo­ked lined angle BAC. the crooked-lined side BA by moving upon the angular point A towards the other crooked lined side CA, will at last coapt and become coincident with the crooked [Page 90] lined side CA then are BA and CA isocliti­cal sides and the angle BAC anisoclitical angle.

17. That by the anisoclitical sides of an angle we understand two such lines, whether the one right & the other crooked, or else both crooked, and whether both of the same or different curvatures, containing a plane angle, so as according to their present site and positure, without inversion of either when crooked, by moving the sides nearer one ano­ther, by the continuance of such motion they can never be brought to be coincident with, and coapted exactly the one unto the other and that by an anisoclitical angle, we understand a plane angle contained under such anisoclitical sides. So all crooked lined angles of concaves, though having sides of like cur­vature, all crooked lined angles of convexes, though having sides of like curvature, all conca­vo-convexe angles, whose sides are of different curvatures, all mixed lined angles, whether re­cto-convexes, or recto-concaves, all mixed croo­ked lined angles whatsoever, are all of them ma­nifestly anisoclitical angles, and their sides ani­soclitical. So in fig. 4th. though AB and AC. be supposed to be of uniform answering and equal curvatures, and likewise in fig. 2d. though AB and AC again supposed to be of uniform, an­swering and equal curvatures, and in fig. 3d. fig, 5, 6, 9, 10, 11. supposing AB and AC not to be of [Page 91] uniform, answering and equal curvatures, by moving the sides AB & AC in that their present fire & positure upon the angular point A, without the inversion of either, when both are crooked, to bring the sides nearer the one to the other, it is manifest that by the conti­nuance of such motion they can never be brought to a coincidence with, and to be coapted exactly the one unto the other, and such also are all angles of Contact whatsoever, as in fig. 7, 8, 11. AB & AC are as uncoin­cidable and uncoaptable as in the former, except only ultradiametral, concavo-convexe an­gles of Contact, which are of equal and an­swering curvatures.

18. That by an Angle of curvature, or coincidence we understand a plane angle, con­tained by any two parts of a crooked line at the point of their concurrences any where to be imagined, or taken in crooked-lines. So in the 7th. fig. let BAD be the circumference of a Circle, Parabola, Hyperbola, Ellipsis &c. At the mean point A, the sides BA and AD contain an Angle of Curvature, or coinci­dence.

19. That by autoclitical curvature, and so by an autoclitical crooked line, we under­stand such a crooked line, as passing from the angular point of a right lined angle, between the two sides, by the inclination of its curva­ture, [Page 92] keepeth the convexe side of its curvature, constantly obverted to one of the sides, and the concave side of its curvature constantly obverted to the other. As in fig. 17. in the right lined angle BAC, the crooked line AEF keeping its convexity constantly obverted to the side AB, and its concavity to the side AC, or so much of it as is intercepted between the intersection at F and the angular point at A the crooked line AEF is autoclitical, i. e. the convexity is all on one side, and the conca­vity all on the other.

20. That by antanaclitical curvature and so by an antanaclitical curve line, we understand such a crooked line, as passing from the an­gular point of a right-lined angle, between the two sides, by the inclination of its curva­ture, hath the convexe part of its curvature, sometimes towards the one side of the right lined angle, and sometimes towards the other side of the right lined angle, i. e. the con­vexeness and the concaveness are not con­stantly on the same several sides. As in fig. 17. in the right lined angle BAC, the crooked line AGH obverting the convexity at G towards AB, and at H towards AC is antanacliti­cal.

These definitions praemised, to give now the true State of the controversy; let there be, as in fig. 12. two equall Circles AHH and ADK [Page 93] touching in the point A: and let AG be the Semidiameter of the circle ADK: and AB be a right line tangent touching both the Cir­cles in the point A: and let AEL be a greater circle then either, touching both the former, and also the right line tangent in the point A: and from the point A draw the right line AC at pleasure, cutting the circle ADK in the point D, and the circle AEL in the point E: now therefore whereas you say that the Recto-con­vexe angles of Contact BAE and BAF are not unequal, and that neither of them is quanti­tative; and that the crooked-lines EA, DA, HA are coincident, Sc. so as to make no An­gle with the right line AB, or one with ano­ther; and that the right-lined right angle BAG is equal to the mixt-lined, i. e. Recto-concave angles of the Semicircles EAG and DAG se­verally, and that those angles of the Semicir­cles EAG and DAG are equal the one unto the other; and that the mixed lined Recto-con­vexe angle FAG is severally equal to all, or any of the former; and that there is no hete­rogeneity amongst plane angles, but that they are all of them of the same sort, and homoge­neal and undevideable into parts specifically different, distinct and heterogeneal in respect of one another, and the whole; and that to any mixt, or crooked lined angle whatsoever that is quantitative it is not impossible to give [Page 94] an equal right lined angle: I acknowledge for all these things you have disputed very sub­tilly; yet I must with a clear and free judgment own and declare a dissent from you in them all, for the reasons to be alleadged in the insuing Discourse.

To clear all which, nothing can be of higher consequence in this Question, then truly to understand the nature of an Angle, what an Angle is, what is an Angle, and what is not. And to exclude the consideration of Angles herein unconcerned; they are plane angles i. e. such as are contained by lines, which lye wholly, and both in the same plane, the dis­quisition of whose nature we are now about. And such is the affinity, which the inclination of one line hath to another in the same plane, with the nature of an Angle, that without it a plane angle cannot be defined, or conceived. For where there is rectitude and voidness of in­clination, as in a right line and its production, there never was justly suspected to be any thing latent of an angular nature; nor between pa­rallels, for want of mutual inclination. But yet the inclination of line upon line in the same plane, is not sufficient to make up the nature of an Angle: for in the same plane one line may have inclination to another, and yet they never meet, nor have in their infinite regular production any possibility of ever meeting; as [Page 95] the circumference of an Ellipsis, or circle to a right line lying wholly without them, without either Section, or Contact; or the Asymp­totes in conjugate Sections, which though e­ver making a closer appropinquation to the circumferences of the conjugate figures, yet in­finitely produced, never attain a concurrency. And though by possibility the inclined lines might meet, yet if they do not, an Angle is not constituted: only there is possibility of an Angle when being produced they shall meet. So as to make up the nature of an Angle two lines must be inclined one to another, and also concurre, or meet to contain on their parts a certain space, or part of the plane between their productions from their point of concurrence, or their angular point. And this being the ge­neral, and proper nature of a plane angle; as it is manifest, that in a right line there is no An­gle, so it is dubious what is to be judged of those curve, or crooked lines, whose curva­ture is either equall, uniform, or regular: Sc. whether in such lines there be not at every mean point an angularness, the lines still lying in the same positure. It is clear in a right line no mean point can be taken, at which the parts of the right line in their present position can be said to have any inclination one to another: that though the parts concurre, yet they want inclination: but in the circumferences of cir­cles, [Page 96] Ellipses, Hyperbola's, Parabola's and such like curve, or crooked lines, being of equal uniform, or regular curvature, no point can be assigned at which the parts in their pre­sent position have not a special inclination one to another. So as, as before, inclination of lines and concurrency making up the nature of an angle; it seems not reasonable to deny angularity at any point of such crooked lines, however their curvature be either uniform, equal, or regular. But I know it will be said that such crooked lines of equal, uniform & regular curvature are but one line, and there­fore, by the definition of an angle, cannot contain an angle; which requires two lines to its constitution. To which I answer, that in like manner, two concurring right lines may be taken for one line continued though not in its rectitude, and then the consideration of angularity between them is excluded: however by reason of the inflexion and inclination at the point of incurvation, there is an aptitude in that one produced line, there to fall, and distinguish it self into two, with inclination of one unto the other, and so to offer the con­stitutive nature of an angle: and so it is at every point of such crooked lines, whose cur­vature is equal, or uniform and regular; being one, or more lines, according as by our con­ceptions they are continued, or distinguished [Page 97] And as a continued rectitude, such as is in right lines, is most inconsistent with the na­ture of an angle, so what should be judged more accepting of the nature of angularity, then curvature is, being thereunto contra­riously opposite, and by all confest most what to be so, saving in the aforesaid cases when curvature is equal, uniform, or regular. But why should equality, uniformity, or regula­rity of curvatures, be so urged in the concern of angularity, by none of which is angularity either promoted, or hindered? For there cannot be greater equality, uniformity, or regularity, then is to be found in Rectitude as well as angles, in right lines as well as Circles; the things constitutive of the nature of an angle, being things quite different from them, Viz. inflexion or inclination and concurrence; which are indifferently found in all curva­tures equal and unequal, uniform, or not uni­form, regular, or irregular: and besides are inseparable from them: curvature and a con­current inflexion, or angularity being but as two notions of the same thing, considered as under several respects: Viz. curvature is the affection of a line considered as one: the same being angularity and inclination, when from any point of the curvature the same line is considered as two: so two concurrent right lines are said to be one crooked line: but [Page 98] when a right-lined angle is said to be contained by them, they are then considered as two from the point of their incurvation, or inclination: and equality, and uniformity, and regularity of curvature implying equality, uniformity and regularity of inflexion, or inclination, it is so far from concluding against angula­rity, that it inferrs it with an additament, Viz. of equal angularity, uniform angularity, and regular angularity; as might be at large declared in the special properties of several crooked lined figures. And in several uniform, and regular crooked lined figures, there are some special points, offering even to view and sense a clear specimen of a more then ordinary angularity, without any such loud calling for the strong operations of the mind; as the vertical points in conjugate figures, in parabola's, and the extream points of either axes, in ellipses, and the like, And in the definition of a plane angle, all the inclina­tion, which is required in the concurring sides, is only, that their inclination be [...] i. e. so as the sides make not one and the same right line, as it is generally understood: howe­ver so far are we from imposing upon any against their judgements, this special sort of angles, that we readily acknowledge, if the words [...], in the definition of a plane angle, be forced from the commonly received [Page 99] sense, of the sides not lying in the same right-line, to signify the non-coincidency of the production of the one side with the other, then according to that gloss upon the de­finition of plane angles, this whole sort of angles is to be rejected; wherein every one is freely left to his own judgement; the diffe­rence being a question and quarrel about words, more then matter, and not concer­ning the present controversy.

However from the whole it out of Contro­versy appears, that recto-concave angles of Con­tact are true angles, contained under two in­clined sides, concurring: also that ultradia­metral concavo-convexe angles of Contact, whether less equal, or greater then two right right-lined angles are true angles, for the sides are inclined and concurr: of their con­currence can be no doubt: and that they are inclined must of necessity be yielded, seeing they neither lye in rectitude, nor which to some might be a causeless scruple, the one in the production of the other: so as their an­gularity is clear beyond all doubt: the incli­nation understood in the definition of an angle being generally any positure of one line with regard to another in the same plane, so as both neither fall in one right line, nor be situated parallel one to another, nor under such a manner of oblique extension, as may [Page 100] render their concurrence impossible: it is not only the oblique positure of one line in refe­rence to another, as contradistinguished from perpendicularness in acute and obsuse incli­nations: but such is the comprehensiveness of its sense, in the present acceptation, that every perpendicularness it self is taken for an inclina­tion.

So likewise it cannot reasonably be denyed but those special angles of Contact, which are the chief subject of this present con­troversy, I mean recto-convexe angles of Con­tact, are truly angles; except either the in­clination of their sides, or their concurrence could be called in question; nothing else being requisite unto the nature, or comprized in the definition of an angle: and the like is to be judged of all other concavo-convexe and con­vexo-convexe angles of Contact. But you said that in the recto-convexe angle of Contact the right-line tangent and circumference make no angle, because the tangent is not incli­ned to the circumference, but coincident with it. What mystery of reason, or force of Argu­ment should be in this deduction? if you say it is coincidence, you mean by non-inclination; I readily yield, where two lines, become coin­cident, their former angle is thereby extinct; as thereby they come under the consideration but of one line: as when two right-lines, or [Page 101] two isoclitical crooked lines, are, one of each of them, so moved about the angular point, till the two lines become one. But where is any such coincidency between right-line tangents and circumferences? or what possibility is there of any such coincidency: a crooked line and a right-line may no doubt be com­mensurate, or proportionable in length, but in position it is impossible: and if we ima­gine the tangent bowed to such a coincidence, then it is not any longer a tangent, or right-line, but a circumference. And though as you urge, a right-line circumduced about any middle point in the side of a regular po­lygone, at last becomes coincident with the side, and looses all inclination and angulari­ty with the side; what doth that concern, or how doth that prove the non-inclination of the tangent to the circumference; your selves sometimes in every point, save only in the point of Contact acknowledging an inclina­tion; and as is else where hinted, in the point of Contact alone, and abstractedly, the incli­nation it self, which is the habitude of the concurring inclined sides, is not to be sought, but only the particular termination of their inclination there. But you will say, it is non-secancy which is meant by this coinci­dency: and what I pray is that more then the Lateran bells to the concern, or consti­tution [Page 102] of angles? Are there not many regu­lar curve lines produced, some infinitely without Section, in which especially the cir­cumferences of circles, you are pleased some­times every where to think you have cause to imagine an angle. It were meet to know the meaning of such odly connext terms, before reasonings upon them be regarded in questions of weight.

That the Contact of the angular sides, is as different from coincidency, as from secancy, is most unquestionably apparent in angles of such sides, as are capable of all three, Viz. Contact, secancy, and coincidence. For example in fig. 16. let there be two Arches of equal Cir­cles DAF and BAC. touching one another in the point A. If the arch DAF be circumduced about the point A as an unmoveable center; at after an infinite succession of secancies, at last all will terminate in a manifest coinciden­cy, and the arch AF be coincident with the arch AB, and the arch AD be coincident with the arch AC. So as secancy, Contact, and coincidency are distinguishable one from ano­ther with as much ease and cleareness, as an odde number from an even. But if it be urged, that you assert not a coincidency between the arch. DAF and the arch BAC, but only that GH being a right line and touching the arch BAC in the point A, that I say the right line [Page 103] GH and the arch BAC are coincident. The vanity of this may be evinced; in that by the same reason it followes, that the right line GH must be coincident with the arch DAF, and so the arch DAF coincident with the arch BAC; the contrary of which is above-shewn and confessed: and besides hereupon should two arches BAC and DAF being con­vexo-convexely posited, and the right line tan­gent GH be all coincident; which I leave for others to say rather then my self.

When a right line tangent and many crooked lines of different curvatures, all touch together in one and the same point, as in fig. 13; you say, though without the angular point of Con­tact the sides are variously divaricated one from another, yet in the point of Contact, they have not several inclinations, for you say they have no inclination at all. The truth is the angular point of its self is not capable of inclinations, nor for the indivisibility of its nature can by any possibility comprehend them: yet that lines concurring in one only point, and presently after receding each from the other, should not be inclined each to other in or at the point of their concurrency, whether it be by Contact, or Section, though in the case of Contacts the inclination be less, then can be expressed by the inclination of any right-line upon a right-line, is absolutely unconceiveable, there being [Page 104] no lineary coincidence, but only of one point between them. For as else where, in the same plane, neither of point to point, nor of point to line, nor of point to plane can be any in­clination: but in the present case of plane an­gles, inclination must be of lines, and may be of them in the very point of concurrence; or else from the point of concurrence they would not part several wayes: for it is their diverse inclination at that point, which makes their departure one from another when they depart from thence. And as even in the angular point of right-lined angles the lines have the same in­clinations as else where, so in all other angles, save only such angles of Contact, as are less then the least right-lined angles, by drawing right-lined tangents to the arch, or arches at the angular point, is shewn in right-lines, either the very inclination of the sides, in, or rather at the angular point, or else the least right-lined inclination, which is greater, or the greatest right-lined inclination, which is less: for though they may differ much in their distance and divarication one from another, without the point of their concurrence; in the point of their concurrence, without much absurdity, they may be said to be equally di­stant, i. e. not at all distant there one from another, for there they are not indeed at all distant any of them from the rest: yet it doth [Page 105] not hence any way follow, that in like man­ner it may be said of them without absurdity, that in the point of concurrence they have equal inclinations, i. e. no inclinations one to another: for though in the point of con­currence it is truth, they have no distance, yet it no wayes as may appear to those that will consider equally follows, that they should there have no inclination.

Besides, that the urging of the coincidency of the sides in recto-convexe angles of Con­tact is most directly opposite to the nature and properties of the special lines under which such angles are contained. For, tis the special propertie of some lines that they can touch, but they can no way be-coincident; as Arches of unequal and unlike curvature: and a right-line and a crooked line: some can be coin­cident, and can no way touch, as right lines: some can both; as Arches of equal and an­swering curvature, which set concavo-con­vexely, and citradiametrally cannot touch, but will fall into a coincidence: but posited convexo-convexely, cannot be coincident, but may construct an angle of Contact.

So as the chimaera of the coincidency of the sides in recto-convexe angles of Contact, if persisted in, is worth laughing at, and like his Philosophy who when every one was at his high Lavolta's, denyed the possibility of [Page 106] motion in the world: But to justify the non-inclination of the sides against the eye and reason, this horridly distorted Monster of their coincidency was introduced. Indeed if they be coincident they make no angle: But it will cramp the understanding of an Oedi­pus to declare how either a right-line, or a crooked-line, touching another crooked-line in one only point and no more, should ever be conceived, notwithstanding to be coincident with the production of the other crooked-line, whether the tangen­cie of the crooked-lines be concavo-con­vexe, or convexo-convexe, i. e. the one within the other, or else the one turned away from the other. But you'll say, you assert coincidency only in the point of Contact. I answer, that's frivolous, not to say ridicu­lous and impertinent; for coincidency in the present question of angles is taken as opposed to inclination, which is an affection and pro­priety of the concurring sides of the angle, not only of the angular point taken by it self abstractedly. As inclination cannot be in a point, though it may be at a point, so a point cannot be said to be inclined unto a line, espe­cially it self being in the same line: it may be said to hold such and such a distance from the line, when it is without it, but not to be inclined unto it. And if the being of the point [Page 107] of Contact, generally as a point of the one side, in the tangent line, as in the other side, whether right, or crooked, make a coinci­dency destroying inclination; then all inclina­tions and angles whatsoever are destroy'd; and every where will be a coincidency, for that is common to every angle to have the an­gular point still common to both sides: and the secant angles might as well be said coin­cident, as the tangent angles, For what you say, that it is tangency as opposed to secancy, that you mean by coincidency; I answer the glosse is improper and besides the anvil and tangency undenied, but in this case imperti­nently by you alleadged, till it be proved that tangency in one only point, and no more, doth quite annul and destroy the inclination of the lines, though on both sides of the Con­tact clearly receding the one from the other: till which be done happiness to my friend and no longer.

I might adde that it is Touching which is only mentioned in the definition of plane angles; but I shall dispatch all by setting the case before you in this Diagramme in fig. 12. if the right line BA touch the arch LA in the point A and the right-line LB be so drawn as to touch the arch LA in the point L, here now is a plane on every side bounded by two right-lines, Viz. BL and BA and one crook­ed-line, [Page 108] Viz. the arch LA. That LBA is an angle will not be doubted: and because the three lines perfectly bound in and limit a plane on all sides; the Arch LA can neither be coincident with the right line LB, nor with the right-line AB, and in the angle LAB, the lines LA and BA; and in the angle ALB, the lines LA and BL concurring without coincidency, and without lying in the same right-line, or the one, so much as, in the production of the other; inclination, and so the true nature of an angle, cannot be denyed them; according to the most severe limitations and hardest glosses, that can with any reason be deduced from the definition of a plane angle. And to make all clear let a Paper, or other plane be cut in the form of the mixed-lined Triangle ALB and the wildness of questioning the angularity of the two recto-convexe angles of contact LAB and BLA will be clear to all persons both rude and learn­ed. I take it therefore for granted; that, all suspicion of coincidency and non-inclination, in whatsoever pertinent sense, of Contact-an­gle sides, being evicted and sent of the scene, all recto-convexe angles of Contact are truly angles.

To passe now unto another of your Thesis's in which you peremptorily conclude Recto-convexe angles of Contact to be devoyd of all [Page 109] quantitativeness; when I urge their quantita­tiveness, I mean not that they can at plea­sure be devided into parts in any given and limited proportion, or by a mathematical ho­mogeneity holding any proportion with the Angle of Contact devided; only that from the angular point between the sides of the least Recto-convexe angle of Contact infinite other lines may be drawn, dividing the angle, though heterogeneally. And certainly its being an an­gle of Contact cannot in the least prejudice its quantitativeness: for it is most apparent that a convexo-convexe angle of Contact contained under Arches of equal curvature, may be di­vided into two mixed-lined, i. e. recto-con­vexe angles of Contact, which hold propor­tion of equality one with another, and each of them is in subduple proportion to the con­vexo-convexe angle of Contact, which was divided: and infinite numbers of Contact-an­gles of several sorts may be adjoyned one unto another, distinct in their situation, without drowning and extinguishing one another and each lying without the other: which is not true of indivisibles when they are adjoyned to, and touch one another. Besides the recto-concave angle of Contact is greater then the greatest right-lined angle, at the angular point of Contact having a manifest inclination of the sides; for concurring, they neither are one [Page 110] right-line, nor one crooked-line. Moreover ultra-diametral concavo-convexe Angles of Contact may be equal to two right right-lined angles, or greater sometimes: As in fig. 12. if the two Circles FAH and ADK be equal and touch in the point A, and AK be dia­meter: and AR be another Arch falling beyond the diameter AK and beyond the right line tangent AB: it is manifest that the con­cavo-convexe ultra-diametral angle of Contact DAH is as to the recesse of the sides equal to two right-lined angles: for the recto-con­cave angle of Contact BAH is equal to two right right-lined angles, deducting the recto-convexe angle of Contact HAS: and the recto-convexe angle of Contact BAD is equal to the recto-convexe angle of contact HAS; therefore the ultradiametral concavo-convexe angle of Contact DAH is equal to two right right-lined angles: and therefore the ultra-diametral concavo-convexe angle of Contact DAR is greater then two right right-lined angles.

And then, what greater Monster is disco­verable in the doctrine of the quantitativeness of the recto-convexe angle of Contact? it is demonstrated that between the right-line tan­gent and the Arch which it toucheth, no right-line can passe: i. e. the recto-convexe angle of Contact is less then the least right-lined [Page 111] angle: but why should hence be inferred that the recto-convexe angle of Contact hath no true quantity? you will say, because a right-lined angle is infinitely divisible into less and less parts, and therefore must at last be less then the recto-convexe angle of Contact, if the recto-convexe angle of Contact be a true an­gle, having truth of quantity. I answer, after all possible divisions of a right-lined angle into parts of its owne kind, I mean such as are made by right lines, the least part is still a right-lined angle; then which the recto-convexe angle of Contact is most fairly demonstrated to be ever less: i. e. the inclination of its sides is eve [...] less: and the convexe arch will still at the angle fall within: so as truly from thence may be inferred, that the recto-convexe angle of Contact can ne­ver be either a right-lined angle, or equal to it, or greater than a right-lin'd angle: but that there­fore it is no angle, and hath no true quantity at all, because it hath not the quantity of a right-lined angle, is a wild and perverse inference, and else-where disproved.

And why should the quantitativeness of the recto-convexe angle of Contact be called in question, because it is demonstrated less then the least right-lined angle? If between the right-lined tangent and the arch a right-line could be drawn, then you would confesse the quantitativeness of it as undeniable: and why [Page 112] doth not the passing of a thousand crooked-lines from the angular point, between the sides of the recto-convexe angle of Contact, as well prove its quantity and divisibility, as the pas­sing of one right-line between them could: there being equal force of proof from the one, as from the other? And when a right-line & two equal Circles, all three touch in the same point; there are two equal recto-convexe angles of Contact adjacent the one continuedly to the other, and situate the one without the other: which in indivisibles is impossible. And when the question is of the quantity of angles, what is it we enquire, but only, what is the inclination of the sides, especially at the angu­lar point? And in recto-convexe angles of Con­tact, the answer is; there is no inclination at all as of a right-line to a right-line, but only as of a crooked-line to a right-line: that it were wildness to say; because in recto-convexe angles of Contact, there cannot be the incli­nation of a right-line to a right-line, that therefore the sides meeting and parting one from another, and not lying both in a right-line, do make no inclination one to ano­ther.

And seeing the convexo-convexe angle of Contact contained under two Arches of equal curvature is dividable into two equal parts by a common right-line tangent to them both; [Page 113] angle of Contact, and of both the recto-con­vexe angles of Contact, clearly appears; though the quantitativenes of every one of them be de­monstrable to be less then the quantity of any the least right-lined angle whatsoever: because right-lines cannot contain a less angle then is agreable to the inclinations they possibly can have one to another: whereas in crooked lines generally, these are the constant properties of curvature; though the angles of coincidence, or curvature, as we for method, brevity and distinction sake named them, may be very unequal one to another, according to the de­grees of the inflexion, inclination and curva­ture; yet constantly the angle on the concave side, from any point of the curvature mani­festly if it be equal, uniform and regular, is still greater then any the greatest acute, right, or obtuse right-lined angle: and the angle between the convexe arch and a right-line tangent, at the same point, is constantly less then any right-lined angle: and yet either may be made still infinitely less, or greater, the recto-convexe angle of Contact still remai­ning less then the least right-lined angle, and the angle of curvature, or coincidence greater still then the greatest right-lined angle: which is as much as to say, that in curvatures the difference between the angle of curvature and two right right lined angles, cannot be a [Page 114] right-lined angle; as in truth it cannot, nor is in reason so to be expected, but of necessity it must be a mixed-lined, Sc. a recto-convexe angle: and is the recto-convexe angle of Con­tact at the same point. And that a recto-con­vexe angle of Contact, by no multiplicity can equal, or exceed a right-lined angle; doth not disprove either its angular nature, or its quantitativeness, both which are otherwise cleared, but it is rather a confirmation of the heterogeneal difference, which is between the angles of the one sort and the other. And in my judgement there needed no greater argu­ment of the quantitativeness of recto-convexe angles of Contact, then the absurdity follo­wing upon the contrary doctrine; that the angle of a semi-Circle, and a right right-lined angle are equal, Viz. The whole and a part, the one being a mixed-lined and the other a right-lined angle; and in the indeavour of coaptation, and being coapted on one side, the other side all the way falls within, or with­out the other, so as both the sides of the one angle are impossible to be coapted to both the sides of the other, but will both still lye within both the sides of the other. And the angles of semi-Circles must either be confessed unequal in unequal Circles, or the curvature of une­qual circumferences, be manifestly against the truth asserted to be equal. And if still they be [Page 115] averred to be equal, its desired their equality should be demonstrated, and the way of ad­measuring their equality shown.

But you will say, if recto-convexe angles of Contact be quantitative, why can they not exhaust any other angle whatsoever contained under the very same sides; for possibly you will urge, that we should not question the homo­geneity between such angles? To this I answer, without examining what homogeneity may in other respects be between them; that a quan­titative and mathematical homogeneity can with no reason between them be imagined: because the difference which is between them is a right-lined angle, to which all angles of Contact whatsoever are heterogeneal; and your self will not assert any mathematical homogeneity, or at least proportionablenes which as to this purpose is all one between any recto-convexe angle of Contact and a right-lined angle. As in fig. 18. let KAB be a recto-convexe angle of Contact, and KAF another recto-convexe angle under the very same sides: and let AD be a right-line tangent upon the arch AF: therefore the recto-convexe angles of Contact DAF and KAB being equall, the right-lined angle KAD is the difference of the recto-convexe angle of Contact KAB and the other recto-convexe angle KAF under the di­varication of the very same sides. So as it is [Page 116] impossible to divide this, or any other angle whatsoever, which is not isoclitical, to divide, I say, all of it into any numbers at pleasure gi­ven of parts, which shall be homogeneal all of them one unto another: for how many soever be homogeneal, the angle of Contact, or that which is taken out of it, or that unto which it adhaeres, will have and make heterogeneity.

And if you say, how can recto-convexe an­gles of Contact be said to be parts of such con­crete and composite angles, if unable by any multiplicity to exhaust the composite angle? omitting the answer that parts are sometimes essential and of the definition, and yet by no multiplicity can equalize the whole; as four angles in the definition of a Tetragone, and a foot line in the definition of a foot Cube: I an­swer chiefly, that where the integral is hete­rogeneal as here, and made up, and properly and naturally resoluble into several heteroge­neal parts, and connot be divided into any num­bers at pleasure of parts all homogeneal, there some parts may never be able by any multipli­city to equalize the whole, or some other he­terogeneal parts. And elss-where that under the same coaptable sides may be angles different in their ultimate kind in some further respects, though not without proportionableness in this instance, is most apparent among right-lined angles, by comparing a right right-lined an­gle, [Page 117] and an acute right-lined angle, and an obtuse right-lined angle together; which all receive their specifick differences from the spe­cifick differences of their inclinations: in right right-lined angles the inclination being no more one way then the other, no more from the angle side, then to the angle side, Sc. perpendicular: in acute angles the inclina­tion being to the angle side: and in obtuse an­gles the inclination being more especially from the angle side.

And yet though we defend the quantitative­ness of recto-convexe angles of contact we are equally obliged to assert their improportiona­bleness to right-lined angles: nor will there be any difficulty in answering that suggession you cast, though in anothers name; that a recto-con­vexe angle of Contact is in proportion less then a right lined angle, as being both homoge­neal; and that by the multiplicity of the recto-convexe angle of Contact, an angle may be made equal to a right lined angle, or greater, only by changing its kind, Sc. into a right-lined angle: Viz. in the same manner, as an acute right-lined angle, being less then a right, or any obtuse right-lined angle, by its multipli­city may with change of its sort and kind be­come equal, or greater then a right, or any given obtuse right-lined angle. To which may be answered, omitting what kind of distin­ction [Page 118] it is which is between acute, right and obtuse right-lined angles, as not pertinent to the present controversy; it would be well done to shew what multiplex of any recto-convexe angle of Contact, is equal to what right-lined angle, that so a right-lined angle might, upon that proportion, be formed equal to, or less then the recto-convexe angle of Contact; con­trary to what has been clearly demonstrated, and is generally by all consented to in Geo­metry. Though acute and right right-lined an­gles are less then any obtuse whatsoever that are right-lined; however they hold propor­tion one with another: but recto-convexe an­gles of Contact cannot be demonstrated to hold any proportion with any right-lined angle, but clearly the contrary. And as by the divari­cations of the sides of an acute right-lined an­gle is made a Genesis of a right and infinite ob­tuse right-lined angle: there cannot so by the divarication of the sides of a recto-convexe an­gle of Contact be effected any Genesis of right-lined angles, but only of recto-convexe mixt-lined angles, whose sides concurre by way of se­ction; and between which and the recto-convexe angle of Contact is no proportion, as there is be­tween the divaricated acute right-lined angle and the other right-lined angles created from that divaricacation. The reason of which is clear, for that the recto-convexe angle of Contact be­ing [Page 119] demonstrated of it self to be less then any right-lined angle whatsoever, by the divarication of the sides of it, there are continually greater and greater right-lined angles added to it, so creating an improportionality between the one, and the other.

The comparison made between Cyphers and angles of Contact, to draw the one, as well as the other into the notion of nullities, is un­happy enough: for single and several Cyphers are not greater, or less in power one then another (though they may make other fi­gures to be so) as the angles of Contact are, and may be made larger, or lesser, and many of them one without or within another, con­tiguously and continuously conjoyned together, with enlargement, or diminution of their an­gularity; which is impossible in indivisibles, and unappliable to, and unintelligible of, mere nullities.

That your Lemma is without exception, and without proof might have been admit­ted: Viz. That two quantities by the ordi­nate application, or motion of a line, or plane, increasing, or decreasing proportiona­bly, whether by a proportionality in the same, or different powers, when the ordinate ap­plication, or motion attains the end, and bounding term of the one quantity, it at the same moment reacheth & hath attained the [Page 120] bounding term of the other, and when it hath passed the one it hath passed the other. But the objection against the quantitativeness of the recto-convexe angle of Contact, which you would hereupon found, hath no reason to expect the like allowance. You say, the right-lined right angle at the point of Contact con­tained under the right-line tangent, and the diameter of a Circle, equally with the circum­ference intercepted, increaseth, or decreaseth by the motion of the diameter upon the point of Contact, as a Center; which is true and acknowledged: and whereas you say, that therefore when the diameter leaves nothing at all of the circumference in its circumvolution about, un-run over, but attains the last bounder and termination of it, in the point of Contact, then as the circumference is quite exhausted and vanished, so is the angle too: this also we acknowledge to be undeniable. But whereas when the diameter is come so near the right-line tangent in its circumvo­lution upon the angular point of Contact, as to intercept nothing at all of the circumference between them, you then imagine still an angle remaining, which you say, is either the recto-convexe angle of Contact, or not less then it; you herein forget the force of your own manifestly true lemma, which you took so much pains to prove; except against Geometri­cal [Page 121] demonstration you could discover a pos­sibility of dividing a recto-convexe angle of Contact by a right-line: for it is out of doubt and in Geometry, as above demonstrated, that when the diameter in such circumvolution intercepts nothing of the circumference between its self and the right-line tangent, the dia­meter is then coincident, and the same right-line with the right-line tangent; and of the for­mer angle therefore, hath lest nothing at all, because of the coincidency of the two right-lines, whose parts can in that positure have no inclination one unto another: and there­fore there is not so much as the least angle of Contact, or any other angle whatsoever, left after this circumvolution: so as the whole Argumentation is a long arrow out of a strong Bow, but quite besides the mark.

It is a seeming weighty objection that which is urged out of opticks, and the usage in that science, to demonstrate in conical figures, the angles of incidence and reflexion to be equal, only with respect to the right-line tan­gent, touching the figure in the point of inci­dence and reflexion, without special respect to the curvature in the conical section. But hereto without wrong, either to truth, nature, or that noble science, may be upon good grounds answered. 1. That opticks is not pure Geome­try, and obstructed stereometry, and mathe­maticks; [Page 122] wherein quantities, mensurations, and proportions are considered merely as in themselves without relation to matter, and the uses to which in other faculties they are appli­cable: but in opticks, is an improvement of what in nature may be observed about lumi­nous and visual beams, and luminous mediums and objects, by mathematical demonstration and assistance. Now 2^ly. Nature doth not tye it self in its wayes so strictly to an indivisible, absolute, vertical punctilio, that if it cannot reach, without impediments, that mathema­tical exquisitenes, it will not act at all. If heavy things on the clive side of an hill cannot descend in a direct line towards the Center, however they will seek it in the nearest oblique line possibly they can, so long as the motion brings them nearer: all animals are not formed in the same mould, or with equal strength of Body, or vigour of spirit. So as in these opti­cal instances, the brave Authors sell not in the least short of their task, when they had de­monstrated the most equal right-lined angles which nature could cast it self into, in the alledged cases, Viz. angles of incidence and reflexion equal, if compared with a right-line tangent at their point. For as elsewhere, they are but very few points in most conical sections, where, by a right-line, their angularity can be divided into equal parts: nature therefore [Page 123] when it cannot attain its prime and general design, which is a perfection and absoluteness in its work, it doth not therefore slugge and do nothing, but seeks to approach its first design as near as possible: and that is in ma­king with right-lined beams equal angles re­spectively to the right-line tangents at the points concerned; which angles so formed at those points with respect unto the right-line tangents, are as is elsewhere shewn, either the least of the greater right-lined an­gles, or the greatest of the less right-lined an­gles: that in such curvatures with great judgement the quality of the angles of inci­dence and reflexion, in beams passing by right-lines, as affected for their directness and short­ness, and as near as possible endeavoured by nature, is in demonstration referred to exami­nation at the right-line tangents of the same points; an absolute equality by right-lines to be made, being most what impossible, and that demonstrably in such curvatures: so as causeless was the exception which was made against the demonstrations of the noble Per­spectivists: nor stood, either they, or nature in need of that improper, lame solution and help, by making recto-convexe angles of Contact to be neither angles, nor quanti­tative.

The truth of the angularity and quantita­veness [Page 124] of recto-convexe angles of Contact thus asserted; however, the rest of the things in this controversy cannot be determined without a clear understanding of what homogenealness it is that is requisite for proportionals, and which is mentioned in the definition of propor­tion. To bring our selves unto the right under­standing of which, know there are three wayes whereby one thing may be said to be greater then another.

1^st. Improportionably, and by the whole kind; as in heterogeneals: in which sometime the least of the one is still, beyond all propor­tion, greater then the greatest of the other: as the least surface is greater then any line; for a longer line may be drawn in the least surface, then any the longest line, that can possibly be given: and so the least body is improportio­nably, and heterogeneally, and by the whole kind greater then any surface: for by the divi­ding, or altering the figuration of the least body, it may be made, by its perimetry, to exceed the quantity of the greatest surface, that possibly can be given: so a point, or a circle of a foot diameter, is said to be less then the whole world; which is to be understood, as by the whole kind, and without proportiona­bleness. And so I doubt not, but it will be con­fessed, that the least recto-convexe angle of Contact, is greater then the greatest acute, [Page 125] right or obtuse right-lined angle: and the least right-lined angle, then the greatest recto-convexe angle of Contact.

2^ly. One thing is greater then another infini­tely, or, if you please, indefinitely: so an infinitely, or indefinitely inlarged line, is longer then any given line, without limi­tation, and so consequenely without propor­tion. Or,

3^ly. One thing is greater then another, rate­ably and according to the proportion, which they hold at the same standard, whereby they are after the same manner measured, in the same kind in which they are compared. So one lenght, or line is longer then another, accor­ding to such a proportion of length when both are measured as lengths at the same standard upon an indefinite line; and this, whether the lengths, or lines be, one, both, all, neither, or none of them directly, or indi­rectly posited, in right, or crooked lines; and whether the one be a depth, and the other a height, or another a breadth, or a fourth a periphery. So one number is greater then another number, according to such a propor­tion: and, though it is well known to be determined in Philosophy, that numbers are of different kinds, yet for the proportion they hold one to another in their common nature, there cannot be denied unto them the truth of [Page 126] a mathematical and analogical homogeneal­ness.

And as things are said to be several wayes one greater then another; so homogeneity wants not its several acceptations. What homo­geneity is, is not at all any where expressely defined in the mathematicks; but we are left at large, rationally to collect, what is by that Term in those faculties to be understood. The most usual acceptation of the word homoge­neity in Philosophy is to compare any divisi­ble being with the parts into which it may be divided: those things being said to be homo­geneal, which cannot be separatingly divided into parts of any other name, or nature then the whole is: as the least separable part of wa­ter is said to be water; of wine, wine: and the least separable part of a line, a line: and those things are said heterogeneal, which by possibi­lity may be separatingly divided otherwise; Viz. into parts of different name and nature from the whole. All which, as appears, hath its dependance upon the similar, or dissimi­lar natures of the whole and parts. So all so­lids, surfaces, lines, and plane angles may be said to be homogeneal; for by dividing and separating them part from part, every part of the solid is a solid, of the surface a surface, of the line a line, and of the plane angle a plane angle: and by taking number in a large sense, every [Page 127] part of a number may be said to be a number. But then if you descend lower in numbers and angles, even and odde numbers, &c. and right-lined and mixed-lined and crooked-li­ned angles will scarce be able to defend their homogeneity at this touch: for Even numbers may be divided into parts that are odde: and a right-lined plane angle may be severed into parts whereof the one may be a recto-concave angle of a semi-Circle, the other a recto-con­vexe angle of Contact; of which last mentioned severed part you say, it is no angle, and there­fore it must be heterogeneal: and we assert it heterogeneal, because it is improportionable, and its quantity not mensurable after the same manner, that the quantity of right-lined angles is measured.

And with reference to the same acceptation, two distinct beings, being compared together, as to homogeneity and heterogeneity, they are then said to be homogeneal, when both having the same name and nature, the one may be the severed part of the other, or both together the severed parts of an integral of the same both name and nature. So all solids are homogeneal: all surfaces: all lines: all plane angles and all numbers whatsoever.

But sometimes also in an abusive accepta­tion, the agreeing, or disagreeing of things in some general, or special kind and nature, [Page 128] passeth for a certain kind of homogeneity or heterogeneity among them: so bodies and surfaces and lines in respect of their common agreeing in quantity, continuity, longitude, latitude and profundity, are partly homo­geneal, partly heterogeneal: so numbers in respect of their general quantitativeness, parity, imparity, symmetry and a symmetry &c. Are partly homogeneal, partly heterogeneal; and so solid, sphaerical, and plane angles, and all their several kinds are partly homogeneal, and partly heterogeneal.

So as from any of the former acceptations it is no way possible to determine what is that mathematical homogeneity which is every where hinted unto us in the definitions of pro­portion; for the bolting forth of which we shall be constrained of necessity to betake our selves to another course, in the quest of which, it is not likely but we shall meet with some abstruseness and difficulty.

To make therefore an Essay; proportion being the habitude of the compared magni­tudes according to quantity, it is an homoge­neity in quantity which is only herein requi­red; not an homogeneity in substance, qua­lity, site, or other kinds and manners of being: for heterogeneity in any of those other re­spects, as substance, quality, site, or other kinds of being, doth not hinder, but they may [Page 129] notwithstanding be one proportionable to ano­ther, if any quantitative homogeneity be to be found amongst them. So the heterogeneity of figure and figuration, which is between a Circle, pentagone, square, triangle, surface of a pyramide, &c. Hinders not, but they may be compared and hold proportion in re­spect of their Area's and superficial contents; wherein they retain a quantitative homoge­neity: and those of them which are meerly bounded planes, also have a homogeneity in respect of their perimetries, or the like quan­tities which they hold in common. So crook­ednes and rectitude are different kinds of posi­ture, and make an heterogeneity in situation; yet a right line and a crooked-line, hold still a quantitative homogeneity, in respect of their longitude and extension. So the different positures of the surfaces in a sphaere, and in a cone, and in a plane, hindereth not their ana­logy in Area and quantity.

But further to pursue this diquisition; though it be homogeneity in quantity, which is herein understood yet it is not the being quantities, or magnitudes in general that doth make up, or can compleat the homogeneity hinted in the definition of proportion: for then lines, and bodies, and numbers, and surfaces, being all of them magnitudes and quantities, there should be a proportionablenes between lines [Page 130] and bodies, bodies and surfaces, surfaces and numbers; which in such a general-homogeneity it were vanity to look for. In like manner, though plane, sphaerical and solid angles be all angles, yet I think there are very few, that because of that their general homogeneity will expect a proportionablenes amongst them.

Besides as little necessity is there of restrai­ning this homogeneity of magnitudes, unto their very last, and in every respect, ultimate kinds: except you will understand it to be in respect of some indefinite, most pure abstra­cted quantity, in which the compared magni­tudes are to be each of them measured. For though some referr all numbers unto quantity, distinguishing them into several kinds, not only from their parities and imparities, with all their variations thereupon, but making every number to be of a different kind, receiving its specifica­tion from the last unite; as some also referr unto quantity all lines, which they distinguish into several kinds, & the like specifick distinctions are made by others of surfaces and bodies: yet between proportionals, for their homogeneity, is not, we confesse, in this sense required, that both magnitudes be of the very same ultimate kind: for hyperbolar, parabolar, elliptical, circular arches, and right-lines are homogeneal in length, extension, and as lines, so holding analogy, though in respect of rectitude and [Page 131] curvature, and their several kinds of curvature they be heterogeneal. So all numbers, even, odde, commensurable, or incommensurable, however, as above is said, heterogeneal, yet as multitudes and numbers in general, and the ac­counts how oft an unite and its parts is posited, or how far in account is proceeded, whether re­trogradely or progressively from an unite; so, all numbers are homogeneal, holding mutual analogy one to another. So notwithstanding the distinction between acute, right, and obtuse, right-lined angles, all truth and exquisitenes of proportion, as the measure of their quantitative relation, is most apparent amongst them.

Which again doth further shew unto us what and how great is the difficulty of limi­ting the homogeneity expressed in the defini­tion of proportion; so as neither to set it use­lesly too high, nor sink it lower then is neces­sary for proportionality. For right right-lined angles are proportionate, one to another, but there can be amongst them no proportion of inequality: and no given acute right-lined angle can have so small a proportion to any acute right-lined angle, as it may have and hath to a right right-lined angle, or to any obtuse right-lined angle: and though all even numbers hold analogy and proportion one to another, yet even numbers are not capable of all diversities of proportion; as in uneven [Page 132] numbers, duple and subduple proportions are impossible: nay numbers in general, I mean unbroken and integral numbers, are not capa­ble of all kinds of analogy; as particularly not of such proportions as are asymmetral.

But not to be nice in my thoughts to my friend, I never in this point and question under­stood any other thing to be meant by homo­geneity in the definition of proportion, then a mensurablenes of the quantities of two, or more magnitudes in the same indefinite quanti­ty for kind, as the measure of their quantity and of the quantity of all their homogeneals, the mensuration therein still being according to that same kind of quantity of which the in­definite quantity is. And so proportion is the rate and habitude, which the rateable magni­tudes hold mutually one to the other in respect of the same way of measuring their quantities, or in respect of the same kind of indefinite quantity in which their quantities are mea­sured. And upon this gloss as the true and ge­nuine meaning of this mathematical homoge­neity, I ever understood that postulate to be founded in which is required and granted, so to multiply any given quantity, as to exceed any other given quantity whatsoever of the same kind. For if that mensurablenes in the same indefinite quantity, as a measure, and according to that same kind of quantity of [Page 133] which the indefinite measure is, were not the very thing designed by mathematical homo­geneity, the matter of the Postulat were not fit to be granted without proof: for it is because they are measured in the same indefinite quan­tity for kind, and according to the same kind of quantity, i. e. they have the very same way of measuring in the same indefinite quantity (which is their homogeneity, by necessity of consequence creating a proportionablenes be­tween them) that the less by multiplying may be made greater then the greater, and the greater by a continual cutting off still more then half may be made less then the less.

And though hereby homogeneity and pro­portionablenes be not made to be the very same thing, however in the mathematicks, where the physical natures of things are not inquired into, the one by a necessary conse­quence doth immediately flow from and is annexed unto the other; and because of their necessary connexion, in usual speech and ac­ceptation, the one may be allowed to be taken for the other. And when in the definition of proportion, proportion is said to be the mu­tual habitude of magnitudes of the same kind according to quantity, or if you please multi­plicity; the meaning is no other, but that pro­portion is the mutual habitude of magnitudes, which have their mensuration after the same [Page 134] manner, according to quantity, or multipli­city; taking the word multiplicity in a large sense: i. e. according to the quantity and mul­tiplicity, which they have each to other in the same indefinite quantity and measure, upon which they are in the same manner and according to the same kind of quantity mea­sured. However in natural Philosophy for very weighty reasons homogeneity and proportio­nablenes are to be acknowledged of very di­stant and different natures. So I presume in Mathematicks it would be taken for a solae­cisme to say a body and a line were homo­geneals and of the same kind, because all se­parable parts of each agree in their being all of them continuous quantities; though in the Physiological school that they do concenter and meet in the same general nature is not de­niable; and so they may carry a seeming shadow of homogeneity, so far as homogeneity may be abusively wrested to denote any such com­mon agreement in a general notion and nature. So it would be a solaecisme in Mathematicks to say, that a solid angle, a sphaerical angle and a plane angle were all homogeneal, be­cause they are all angles and every separable part of each is an angle: but to pro [...]e Mathe­matical homogeneity, the mensurablenes of the quantity of the compared magnitudes in the same indefinite quantity, or measure for [Page 135] kind, and according to the same kind of quan­tity with the indefinite measure is to be made out: for that all are quantities, or all angles, makes them not in the mathematical school homogeneal; except by reason of this men­surability of the quantity of both in the same indefinite quantity, or measure, according to the same kind of quantity, the less by multi­plying can be made greater then the greater, and the greater by dividing less then the less. And indeed this is the true homogeneity, not denoting a general conveniency in their natures in respect of some abstracted notion, but ra­ther a special identity by reason of their men­surablenes in the same substrate kind of quan­tity and measure, only with difference, or proportionablenes of magnitude between both the wholes, and all the least, or greatest, pro­per, i. e. homogeneal parts of each: as if one be a line, so is the other, and the greatest and least proper, i. e. homogeneal or homome­tral parts of each are lines and proportionable to either. For, if besides the mensurablenes of the quantity of the compared magnitudes in the same indefinite quantity, or measure, be not also added that condition, that in that indefinite quantity, or measure they are also mensurable according to the same quantity for kind, of which the indefinite measure is, and so consequently proportionable one to another; [Page 136] by nothing will it yet be determinable, whether all angles be not homogeneal: for in every angle, though of several kinds, every part of each angle is an angle: nor will it be deter­minable, whether all continuous quantities be not homogeneal; for every part even of hete­rogeneal continuous quantities is a continuous quantity: nor will it be determinable whether all numbers be not homogeneal; for that all numbers are of the same kind will be found a doctrine of very hard digestion any where save in the mathematick school: yet accor­ding to this explication of mathematical ho­mogeneity, notwithstanding the diversity of their kinds in other philosophical considera­tions, they have in them a clear mathema­tical homogeneity: and even an unite, which in other parts of Philosophy is not passable for a number, will fall also within the verge of the same homogeneity; as will also all the parts of an unite, whether commensurable, or incommensurable. And this explication of mathematical homogeneity will be allowed its due right and justification more easily, by those who note how the main matter and de­sign of mathematical definitions is but exege­tical to clear up what is meant by the terms in those sciences used: for what other occa­sion could there be in the Mathematicks to intermeddle with homogeneity, but to ex­plain [Page 137] the noble points of proportion and pro­portionality.

And yet though in mathematicks there be such a grand affinity between the proportio­nablenes and homogeneity of magnitudes and in common use and spee [...]h the one may be put for the other, yet as above the notions are easily distinguishable by the understanding: Viz. two, or more magnitudes are said homogeneal chiefly in respect of the same way of measuring them, or in respect of the same kind of inde­finite quantity in which they are both men­surable: but they are said proportionable in respect of the mutual habitude and quantita­tive relation which is between themselves upon such their mensuration, in the same way, or according to the same kind of quantity. So all finite lines are homogeneal, as mensurable in the same indefinite line; but that one finite line is double to another is the habitude of the one to the other declared upon that men­suration.

That upon the whole matter mathematical homogeneity doth not insist only upon iden­tity in kind at large, or restrain unto iden­tity in kind at all points, and in every re­spect and consideration; but it is identity in quantity and therein particularly in the man­ner of the mensuration of their quantities, in which the homogeneity of magnitudes is chiefly [Page 138] lodged: and that is that which I ever under­stood in those words (according to quantity, or if you please, rather quotuplicity) which are in the definition of propo [...]tion, viz. quan­tity and quotuplicity in the same way of mea­suring.

And if you urge here, according to this account I must conclude all incommensura­bles to be heterogeneal; I answer, not in the least: for though they may have no common measure, which can by possibility exactly mea­sure both or all, yet there may be a common indefinite measure, in which each may exactly measure forth its own quantity. As for example in fig. 24. Let A and B be incommensurable lines, and DC. an indefinite line, beginning at D, and on the part of C infinitely produ­ced: from the point D in the line DC take the line DE equal to the line A: and also in the line DC take the line DF equal to the line B. here in the line DC as a common in­definite measure of their homogeneity, the two lines A and B, though incommensurable, have measured forth themselves by the lines DE and DF. So in fig. 1. upon A the angular point of the right-lined angle BAC, as Cen­ter, draw the Arch dfe cutting the line AB in the point d, and the line AC in the point f, and the right-line Ae in the point e, so as the Arch df be incommensurable to the [Page 139] Arch fe. Here the two right-lined angles dAf and fAe are incommensurable, yet have a common way of measuring their quan­tity and proportion, Viz. by Arches of Circles drawn upon the angular point, as center, intercepted respectively between the sides: as by the Arch dfe, the one being in that Arch measured by the Arch, df, the other by the Arch fe. So as by mathematical ho­mogeneity is understood an homometricalnes, or autometricalnes with the necessary conse­quent of a rateablenes therein, without any necessity of symmetricalnes between themselves at all. For still, if in homogeneity besides ho­mometricalnes in the same indefinite quantity be not also included, that the mensuration of both be according to the same kind of quan­tity, of which the indefinite quantity, in which they are measured, is; a point and a line may measure themselves in an indefinite line, and a line and a surface may measure themselves in an indefinite surface, and a surface and a solid may measure themselves in an indefinite solid. But true mathematical homogeneity is when two, or more quanti­ties being mensurable in the same indefinite quantity, or measure, and according to the kind of the indefinite quantity are by conse­quence rateably, i. e. proportionably and ho­mometrically equal, or the one bigger then [Page 140] the other, not the one infinitely bigger than the other, as an infinite line is longer then a finite line, for between such it is acknow­ledged there is no proportion nor mathema­tical homogeneity; nor the one bigger then the other by the whole kind, as every solid is bigger then any surface, and every right-lined angle then any recto-convexe angle of Contact: and proportion is the rate, or quantity of their mutual habitude in that their homogeneity, or homometricalnes: i. e. proportion is the rate, quantity, or account of their proportionality. And applying our minds rightly to conceive of homogeneity as mathematical; of necessity such, and no other can be the notion of it, thereby making two quantities to be rateably, i. e. proportionably equal, or the one bigger then the other, and according to their capacity and possibility in any rate and proportion so to be constituted and set out. For the genuine well known notion of homogeneity in general, what is it but that all and each of the proper, i. e. ho­mogeneal parts, and the whole, fall under the same Denomination and nature, as if one be a line, all the rest to be lines, if one be water, or stone, or oyle, &c. all the rest of the parts and the whole to be so also? And in the mathema­ticks what is said properly and homogeneally to be a part of any magnitude, but only such lesser magnitudes separable from it as are able to mea­sure [Page 141] out, i. e. by their multiple to exhaust the first magnitude? For a surface is not said to be an homogeneally proper part of a solid: nor a line of a surface. And by such measuring forth of the integral magnitude by its homogeneally proper, i. e. exhausting parts, is not understood the sym­metry of the parts, and whole, for the whole and its true homogeneal parts, may be incom­mensurable; but that which is understood is the true mathematical homogeneity of the whole and the parts in respect of their com­mon way of measuring, and the proportiona­lity, which is thereupon lodged between them So as by laying these undeniable remarks to­gether, mathematical homogeneity in respect of the same integral magnitude, is that every part thereof being proper, i. e. mensurable in the same indefinite measure according to the special kind of its quantity, and therefore able to exhaust the whole, be all of them only in respect of their quantitativenes, not their figu­ration, or other respects, of the same nature and denomination, and any proportion what­soever (according to the capacity of such kind of quantities) possible to be constituted amongst them. And by consequence several magni­tudes are then said to be mathematically homo­geneal, when being the one able to exhaust the other, & so in proportion one unto ano­ther, both and all the proper parts of each, [Page 142] all of them, as being of the same nature fal [...] under the same quantitative denomination.

And here it is worth our noting, how some magnitudes are so homogeneal, that they can­not be separatingly divided into parts, or magnitudes which are heterogeneal; but all the parts into which they are separatingly di­vided will still be homogeneal, both one with another and with the whole: So a line can no way be separatingly divided into parts which are magnitudes, but each is a line ho­mogeneal to the whole and to all parts what­soever of whatsoever other line: neither can a body and surface be separatingly divided, but into bodies and surfaces; each of whose parts are still able to exhaust the whole and are thereunto proportionable. But such an ab­solute homogeneity is not in all other magni­tudes, especially in heterogeneally concrete magnitudes, but that the whole may be di­videable into parts separable one from another, which yet are heterogeneal: as in the present case of angles a right right-lined angle is di­videable into a recto-convexe angle of Contact, and the recto-concave angle of the semi Cir­cle; which are separable one from another, and therefore truly parts; yet neither of them is homogeneal to the whole, nor one of them unto the other: for neither can the whole right right-lined angle, nor the recto-concave angle [Page 143] of the semi Circle ever be exhausted by any number whatsoever of such heterogeneal parts, as is the recto-convexe angle of Contact: nor ever any equality, or other proportion can possibly be shewn between the right right-lined angle, and the recto-concave angle of the semi Circle, because there is no way pos­sible in which their quantities can be propor­tionably mensurable. For not without very good reason unto all magnitudes are to be allowed their special properties; as to all po­situres and figuration, theirs. To angles these things are peculiar, being otherwise in other magnitudes: Viz. in angles which are truly and on all hands confessedly homogeneal, you cannot to any given angle, set forth another of the same kind in any given proportion at pleasure: for every right-lined angle by a ne­cessity of nature, must be less then two right right-lined angles: and in a plane all the an­gularity at any point cannot exceed what the circumjacent space, or plane is capable of; which is only four right right-lined angles. That as number cannot be infinitely divided without fraction; so angularity cannot at plea­sure at the same point, in the same plane, be inlarged: whereas some other quantities have both infinite divisibility and infinite multipli­cability.

So another property of the magnitude of an­gles [Page 144] is, that it may not only in notion and spe­culation, but in truth and severingly, be di­vided into parts either able, or unable to ex­haust the whole: as when a right right-lined angle is divided into the recto-concave angle of a semi Circle, and a recto-convexe angle of Contact: you may sever them the one from the other: and angularity is equally, if not much more apparent in the recto-convexe an­gle of Contact, then in the recto-concave an­gle of the semi Circle; yet the one of them is demonstrated and confessed unable ever to ex­haust the right right-lined angle, the other not.

A further property of the magnitude of an­gles is, that sometimes the same part, which hath already been severed from it, cannot ex­actly and immediately again by its equal be severed from it on the same side, though the remaining angle be by the whole kind greater. So after a recto-convexe angle of Contact is taken out of a right right-lined angle, there cannot again immediately on the same side, be severed from the remaining angle another angle equal to the recto-convexe angle of Con­tact, which was before severed from it. If it can, let it be performed.

Also the divisibility which is in the magni­tude of all Angles, though boundles and in­finite, in some however, leaves the dividing [Page 145] of the Angle into two equal parts, impossible: as notwithstanding the perpetual divisibility of lines, the side and diameter of a square are left incommensurable. So some other angles may be divided into two equal parts, but it is impossible to divide them into three equal parts: as convexo-convexe angles of Contact, with infinite other convexo-convexe angles and con­cavo-concave angles being contained under equal, uniform and answerable Arches.

To consigne this point, the principal thing we have laboured herein to dilucidate, & as we doubt not have effected is, that mathema­tical homogenealness is not an homogeneity of all the parts whatsoever, that are in the magni­tudes, which are homogeneal in respect of some special way of measuring their quantities; or an undivideableness of such homogeneal magnitudes into parts otherwise heterogeneal according to which acceptation the word is chiefly taken in other parts of Philoso­phy: for there is no right-lined angle whatsoever, nor any other angle whatsoe­ver, but as is up and down herein shewn may be separatingly divided into heteroge­neal parts: but mathematical homogeneity is homogeneity in the way of measuring the quantity of the compared magnitudes, Sc. in the same indefinite measure and quan­tity, and according to the kind of the [Page 146] indefinite measure, and which thereupon fol­lows, a proportionality between them, in respect of their common way of measuring: and of this mathematical homogeneity, fair foot-steppings are to be found every where in the deducing of those demonstrations which con­cern proportions and proportionals. That, such magnitudes as have no common way of measuring their quantity, as weights and mea­sures, are heterogeneal: or if they have a com­mon way of measuring in which they may measure themselves, but therein do not mea­sure themselves according to the same kind of quantity with the indefinite measure, and so want proportionality, yet notwithstanding they are heterogeneal: as all recto-convexe an­gles of Contact, all recto-concave angles of semi Circles, all recto-convexe angles of semi Circles, all acute, or right right-lined angles, these may all measure themselves, and in what order their sides fall within, or without in any obtuse right-lined angle whatsoever: yet because this their homometry is only of the situation, or order, in which the sides part from the angular point, but not of their quan­tity in an indefinite measure and according to the denomination of the same quantitative mea­sure, so as to lodge a proportionality between the magnitudes so compared together in their common way of measuring; they are not, nor [Page 147] can thereby be vindicated from their other­wise innate mathematical heterogeneity: which concerning some of them is confessed on all hands and is without the verge of the controversy. And as follows, angles are of a concrete nature, having in them something quantitative and something not quantitative: whereas that which is to be the indefinite measure of homogeneal quantities is to be con­sidered abstractly as quantity without hetero­geneal concretion: so it is the circumference of a Circle that measures all right-lined angles. And when all plane angles are said to be ho­mogeneal, it is not in respect of a common in­definite quantity by which they are all mea­sured, which the recto-convexe angles of Contact doe sufficiently evince, but as is ma­nifest it is only because of the position and si­tuation of the sides in the same plane; which homogeneity is of no concern unto quantity, nor by any necessity can thereupon infer the consequent of proportionableness.

But to proceed; as is said, besides the former mathematical and quantitative homogeneity and heterogeneity there is also an extramathema­tical and extra-quantitative homogenealnes and heterogenealnes in angles every where observa­ble in their shapes, figures, positure of their sides, such like schematismes and other respects. In general as is above hinted, every part of a [Page 148] plane angle is a plane angle, even the recto-convexe angle of Contact, however you deny it to be an angle and quantitative: but then this is not a mathematical homogeneity, but only in respect of a certain figuration, in re­spect of the positure and situation of the sur­face in which those angles are; shewing how all plane angles from the greatest to the least, agree in that particular of their general figura­tion, Viz. of having their containing sides to lye still in the same plane; whereby they di­stinguish themselves from all other superficial angles, which are heterepipedal, whose con­taining lines, or sides lye in several planes: such as are all sorts of sphaerical, cylindrical and conical surface-angles. But if ever a mathe­matical and quantitative homogeneity be pro­ved among all plane angles, you that know that it is not my use to start from my word, shall hereby rest assured, upon the first sum­mons I will give up this cause.

And we are not to think strange, that a figuration is asserted to be in angles; for if we seriously consider, wee shall find there is shape and figure in angles, as well as quantity; as lines, and surfaces, and bodies have their fi­gurations, the positure of their parts, their shapes and forms, as well as their quantities and magnitudes: in each, their figuration being manifest; Viz. in lines, in respect of [Page 149] their lineary positure: in surfaces, in respect of their superficial positure: in bodies in re­spect of their solid positure: and in the casting of each of their schematismes quantity is invol­ved, as length, breadth, depth, Viz. their quantities, and the quantities which are com­pounded of several, or all of them together.

And here by the figuration which we assert in angles, we cannot be thought to mean that any right-lined figure can be compleated, perfectly to bound up a plane on all sides, by one angle; it being beyond the power of two: and three, being the least number of angles, requisite so to constitute and perfect­ly limite out a right-lined plane figure. And though some plane figures are perfected, and perfectly bounded, without any such angles as are contained by sides concur­ring by way either of section, or Contact, as namly all Circles and Ellipses; yet the angularity of curve coincidence is every where found, or at pleasure assignable in the boun­ders of such figures. But our meaning is, a plane angle, though most what it do not by the continuation and production of its sides, per­fectly bound in and limit out a certain plane and space on evry side, however being the mutual habitude of concurring lines, it gives an imperfect figuration to the plane and space on its part. And as a bounded plane cannot be [Page 150] without some kind of plane figure, so a limi­ted angle ever implies in it an imperfect figu­ration of some sort, or other. For figuration is the consectary of material finiteness and li­mitation in the position of lengths, breadths, depths, surfaces and solidities; that every angle having its limits and bounds cannot be thereof destitute. And if the name of figures be so frequently given to hyperbola's, parabola's, and the like, which neither do, nor ever can by any possible production perfectly bound in their planes; what reason is there then why angles should be denied an imperfect interest in the name. Besides as a plane in its own ge­neral nature at large doth not denote any spe­cial plane figure; but the rise of figures, I mean plane figures, is from the bounding of the plane: so it is in angles, as they by the mutual habitude of their concurring sides give imperfect limits and bounds unto the space and plane, so they therein make an imperfect figuration. That in angles something of form and figure is to be noted as well as magnitude. And one line cannot concurre with and be inclined upon another, but an imperfect fi­guration will arise from that their mutual in­clination. And the same two angles may have the inclination, i. e. the recesse of their re­spective sides one from another equal, though there be no analogy between the figurations [Page 151] of the angles, or the shapes in the which the sides are inclined in the one and in the other. For by reason that in angles form and figure are to be observed as well as quantity; crook­ed-lined and right-lined angles may be equal in some particular quantity, yet other-wise not of the same kind: they having equality in some magnitude, but being distinct in the man­ner of their forming, figuration and constitu­tion: as equality may be between a square and a triangle, though figures altogether dif­ferent in kind.

And in respect of such their figurations, plane angles receive distinction, either from the diverse manners in which their contai­ning sides do concurre, or else from the diverse natures, and figurations of the lines under which they are contained, or, which is tantamount from the diversity of the inclinations and inflexions, or rather inclina­blenesses and imflexiblenesses by which they are inclined each to other, or from several of these grounds of distinction taken together.

Plane angles from the different manner of their sides concurring, may aptly be thus dis­tinguished, Viz. into angles, whose sides con­curre by way of section; or else that have their sides concurring only by way of touch in some single singular point without mutual section; or else their concurrence is in curvature, where, [Page 152] after the meeting of their sides in the angular point, the sides do not in their productions depart one from another, neither by way of touch, nor section; but become, the produ­ction of the one side coincident with the other side; so as this kind of angles may aptly be called angles of coincidency, or angles of curvature: and in these lies the genuine Ratio and true account of the curvature.

From the diverse figuration of the lines, under which a plane angle is contained, very many differences of angles may arise, accor­ding to the various distinctions, of which lines themselves are capable, I mean such lines, as fall not without the capacity and compre­hensivenes of the same plane: as that some are helicoidal, some parabolar, some ellipti­tical, &c. But as of lines, so hence of angles, the chief and primary distinctions are espe­cially these, Viz. that plane angles are either right-lined angles, contained under two in­clined right-lines; or not right-lined angles.

Not right-lined angles, are either mixed lined angles; contained under one right-line, and one crooked-line: or crooked-lined an­gles contained under two crooked-lined sides. And from the several kinds of special, or or­dinary curvatures, as Circular, elliptical, hy­perbolar, &c. The mixt-lined, and crooked-lined angles, are capable of many farther and [Page 153] more particular distinctions; but especially from the site of the convexeness, or concave­ness of the lines to, or from the angle side: though all such secondary distinctions rising from these two last mentioned heads, are as properly and pertinently referable to the other ground of distinguishing plane angles, taken from the differences which may be in the in­clinations of the one containing side to the other. For a vast difference is in the incli­nation of a crooked-line, by obverting the con­cave, or convexe side to any other line. So the constituting a circular, or elliptical arch &c. For one side, makes a vast difference in the inclinations, because of the difference in their curvatures.

Also another principal distinction of angles from their sides, may be into angles, whose sides are coaptable, and by possibility may be coincident one with another: or else such as have between them no possibility of coapta­tion and coincidence. Of the former sort are all right-lined angles, and all concavo-convexe angles contained by Arches of equal homoge­neal, uniform, regular, or answering curvatures: of the later sort are all other; whether mixt lined angles, or crooked-lined angles; whether they be mixed crooked-lined angles, or unmixt crooked-lined angles. And consequently there­upon, besides the numerous distinctions of an­gles [Page 154] in respect of their different inclinations, such as above mentioned; one is more emi­nently material above the rest, that the incli­nation of the sides, is sometimes with an equa­bility all along their production; though ima­gined to be infinitely extended, in such lines as by possibility may with reason be imagined so to be: and sometimes there is nothing of equability to be found in the inclination of the several parts of each side to the other; though it may be one of the sides be a right-line, or an Arch of most equal, uniform, re­gular and homogeneal curvature. And this equability and inequability of the inclination of the sides, strangely alters the properties of angles. As in right-lined angles, for the equa­bility of the inclination of the sides, no parts of the one side are more inclined then the rest unto the other side: and so in concavo-con­vexe angles of equal curvatures, no parts of the one Arch are more inclined then the rest unto the other; but the one Arch is all along inclined unto the other, as at the angular point; and the inclination, which the one bears unto the other at the angular point, is obviously expressable as to the quantity of the recesses, which they make one from another, by the inclination of a right-line to a right-line, except when the inclination of the Arches is equal to, or greater then two right right-lined [Page 155] angles. And in such crooked-lined angles, whose sides have equability of inclination, the points, which from the angular point are at equal di­stances along the Arches, are also absolutely at equal distance from the angular point, along the chords: and right-lined tangents at any two such homologal points, where ever taken, al­wayes meet and contain a right-lined angle, equal as to the quantities of the recesses of the sides, to the crooked-lined angle contained by the two Arches; as is obvious to demonstrate, especially in circular Arches. And the right-lined angle contained under the two right-lined tangents touching at the two homologal and answering points, which is equal to the iso­clitical crooked-lined angle, if the two right-lined tangents occurre on that side of the right-line connecting the homologal points on which the isoclitical angle falls, then it is the angle contained by the two right-lined tangents into whose space part of the space comprized be­tween the two Arches at first falls, which is equal to the crooked-lined isoclitical angle: but if they occurre on the other side of the right-line connecting the two homologal points, i. e. aversely from the crooked-line isoclitical angle, then it is the complement of such an angle, which is equal to the crooked-line iso­clitical angle: but if the two right-lined tan­gents occurre in one of the homologal points, [Page 156] the angles either way contained under the two right line tangents are equal, viz. right right-lined angles; either of them making forth what is herein asserted. As in fig: 19. un­der the two circular isoclitical arches bda and acn let there be constituted the isoclitical angle bac; and let the right-line ag touch the Arch acn in the point a: and let the right-line af touch the Arch adb in the point a: so making the right-lined angle fag equal to the isoclitical concavo-convexe angle bac. Then take in the Arch adb any point at pleasure, Viz. the point d: and draw the chord ad. Then in the Arch acn take the Arch ac subtended by the chord ac equal to the chord ad. Therefore because of the iso­cliticalness of the circular Arches the two points d and c are two homologal, i. e. answering points the one in the one Arch, the other in the other, Viz. the point d in the Arch adb and the point c in the Arch acn. Then draw the right-line dc connecting the two homo­logal points d and c. Also draw the right-line de touching the arch adb in the point d; and the right-line ce touching the Arch acn in the point c. And let de and ce the two right-line tangents be produced till they meet in the point e; which in this figure is on that side of the right-line dc on which the con­cavo-convexe angle cab lyeth. I say there­fore [Page 157] that the right-lined angle dec contained under the two right-lined tangents de and ce touching the Arches respectively at the homo­logal points d and c is equal to the right-lined angle fag, contained under the two right-line tangents fa and ga touching the Arches re­spectively at a the angular point of the isocli­tical concavo-convexe angle. For the right-lined tangent fa cutting de the other right-lined tangent of the same Arch adb in the point h; and the right-lined tangent de of the Arch adb cutting the chord ac in the point k; upon this construction the right-line da is equal to the right-line ac; and the right-line tangent dh is equal to the right-line tangent ha: therefore the right-lined angle adh is equal to the right-lined angle dah and so to the right-lined angles ace and cag severally. And therefore the right-lined angle ahe being equal to the two right-lined angles hda and dah taken together, and the right-lined angle hda being equal to the right-lined angle cag; the right-lined an­gle ahe is equal to the two right-lined an­gles cag and dah taken together. Therefore that which maketh each equal to two right right-lined angles; the two right-lined angles hka and hak taken together are equal to the two right-lined angles hak and dal ta­ken together. Therefore the right-lined angle [Page 158] hka is equal to the right-lined angle dal. Therefore the right-lined angle cke is equal to the right-lined angle dal. And therefore that which makes either equal to two right right-lined angles, the two right-lined angles kce and kec together taken are equal to the right-lined angle dag, which is equal to the two right-lined angles dac and cag taken together: and the right-lined angle cag is equal to the right-lined angle kce: there­fore the right-lined angle kec is equal to the right-lined angle dac: therefore because the right-lined angles dah and cag are e­qual; also the right-lined angle kec Sc. dec is equal to the right-lined angle hag Sc. fag, which was to be demonstrated.

But if the two right-lined tangents de and ce as in fig. 20. do not occurre towards the concavo-convexe angle bac, but on the other side of the right-line dc in the point e; then is the right lined angle dec contained under the two right line tangents de and ce touch­ing at the homologal points d and c, not equal to the concavo-convexe isoclitical angle bac, or the right lined angle, equal unto it, fag, but to its complement unto two right right-lined angles, Viz. unto the right lined angle fal; the right line la being the production of the right line ga. For as before by constru­ction, the right line chords da and ca to the [Page 159] homologal points d and c are equal: and the right line fa cutting the right line de produced in the point h, the right lines dh and ah being two right lines tangents of the same Circle adb, oc­curring, are equal. And let the right line ac pro­duced, occurre with the right line de produced, in the point k. As appears; the right lined angles adh, dah and cag, as before, are equal; and the right lined angle ahe is equal to two right right lined angles, all but the two right li­ned angles hda and had that is, all but the two right lined angles cag and had. Therefore the right lined angle ahe being equal to the two right lined angles hka and kah; the two right lined angles hka and kah are equal to two right right lined angles all but the two right lined angles cag and had. Therefore two right right-lined angles are equal to the four right lined angles cag and had and hka and kah. Therefore out of equals ta­king equals, the right lined angle hka, which is the right lined angle cke is equal to the right lined angle dal. Therefore what on either side remains to make up two right right lined angles on either part; the two right lined angles kce and kec are toge­ther equal to the right lined angle dag, which is equal to the two right lined angles dac and cag taken together. And producing the right line ec till it cut the right line ag in the [Page 160] point g; the right lined angles cag and acg and kce are equal. Therefore the right lined angle kec is equal to the right lined angle dac. And because the right-lined angle dah and cag are equal, therefore adding the common angle fac, the two right lined angles dac and hag are equal. And therefore the right lined angle kec is equal to the right lined angle hag. And therefore their comple­ments unto two right right lined angles, Viz. the two right lined angles dec and hal are equal; which was to be demonstrated.

So in fig. 22. if the right line tangent de passe directly unto the other homologal point c, as it doth when the isoclitical concavo-con­vexe angle is equal to a right right lined an­gle, and the homologal points d and c are taken at quadrantal or other distances from the angular point; then most manifestly the right­lined angles dcg and fag are equal, being under the two and two respective right lined tangents; and between the angles and their complements unto two right right lined angles is no difference, as appears by what is in the former demonstrations. So in fig. 23. if the homologal points d and c be so taken, that the chords da and ca be the diameters: then producing the right line tangent ce till it oc­curre with the other right line tangent de in the point e, and with the other right-line tan­gent [Page 161] fa in the point f; the right lined angles cfa and fag are equal. And in the trape­zium def a the two right lined angles eda and daf are each of them a right right-lined angle, therefore the right-lined angle dec is the complement of the right lined angle efa unto two right right-lined angles, i. e. it is the complement of the angle fag unto two right right lined angles; which was to be de­monstrated. And like demonstrations may be formed upon every other case. But where the isoclitical arches are not circular the demon­strations must vary according to the propriety of every several curvature, notwithstanding its equability and isocliticalness.

But to return, in angles contained by sides whose inclination each to other is without any such equability, (except only in unmixt crooked-lined anisoclitical angles) the account of the homologal points along the chords and along the arches is still different: but in none of them do the right line tangents from the two and two homologal points, still in their meeting make the two same angles which are made by the two right lined tangents at the angular point; nor can any right line angle ex­presse the inclination, which the sides have each to other at the angular point.

Hence is manifest how the same angle may from several of the grounds of distinction here [Page 162] proposed be referable to several heads, or kinds. So right lined angles as isoclitical, have al­wayes equability of inclination; and the con­currence of their sides is alwayes by way of section and cannot be by Contact, or coin­cidency. So mixed lined angles being anisocli­tical have always inequability in the inclina­tion of their sides; and their concurrence may be either by Contact, or Section, but never by coincidence. And in crooked lined angles, their sides may have either equability, or ine­quability of inclination: and accordingly the concurrence of the sides may be by section, or Contact; and with, or without possibility of coincidence Sc. Isoclitical crooked-lines, but they must be posited convexo-convexely, or anisoclitical crooked-lines posited whether con­vexo-convexely, or concavo-convexely, or a right-line and a crooked line, any two of these may touch without cutting, and so the angular sides have inclination tangent and not secant, So isoclitical concavo-convexe angles may have sides circumducted to coincidence: but the same sides posited concavo-concavely, or convexo-convexely become anisoclitical in the circumduction, one in respect of the other; yet either is isoclitical sometimes and in some cases with the production of the other.

From these things though distinguishing an­gles into their several kinds, only with respect [Page 163] to the diversity of their figuration, may how­ever more abundantly appear how unma­nageable a task they take upon themselves, who to exclude recto-convexe angles of Con­tact from being angles, and from quantitave­ness, would force all plane angles to be of the same kind, allowing no specifick difference possible among them. Not here to pursue what other diversities in kind may be observed among angles; how can the inclination of a crooked-line upon a right-line differ less then in kind from the inclination of a right-line upon a right-line? For as a right-line and a crooked-line agree as lenghts and in lineari­ness and are therein mathematically homoge­neal, but as right and crooked differ in kind and have therein heterogeneal figuration; the crookedness of the crooked-line having no analogy to the rectitude of the right-line, nor the rectitude of the right line any proportio­nableness unto the curvature of the crooked-line; so the inclination of a crooked line upon a right line, and of a right line upon a right line they agree in the common nature of incli­nation, and the one may be greater and the other less as the respective sides fall within, or without: but this relation of greater, or less is without any proportionableness, and only by the whole kind, depending upon the passing of the respective sides, the one within, or [Page 164] without the other. For these two inclinations differ so far in kind that neither the curvatu­re of the mixt lined inclination hath any thing in it conform, or proportionable to the re­ctitude of the right-lined inclination, nor the rectitude of the right-lined inclina­tion to the curvature of the mixt lined inclina­tion. In a word, so different is the inclinable­ness of a crooked line upon a right line, from the inclinableness of a right line upon a right line, that it is impossible for the one ever to be either equal, or any way determinately pro­portionable unto the other: because the coap­tation of a right line as a right line, to a crook­ed-line, as a crooked Iine is against the proper­ties of their figurations, kinds and natures. And for what reason should there be lesser difference between a crooked-lined inclination and a right lined inclination, then there is between a crook­ed-line and a right line? Yet all this their di­stinctness concerns only an heterogeneity in their figuration and not at all, or not primarily their quantities.

The argument, if they be angles, or plane angles, they are homogeneal and of the same kind, is of no more force then this consequence; if they be quantities, or continuous magni­tudes, they are homogeneal and of the same kind. And they that deny all heterogeneity in angles, because they are all angles; will find [Page 165] it an hard task upon the same ground to main­tain an analogous homogeneity, or any other considerable homogeneity, between right lined angles and sphaerical angles, or any other an­gles, made by planes cutting the heterepipe­dal surfaces of solids, and especially solid an­gles, of what sort soever. And to yield that all plane angles are homogeneal; for it is true: and the most absolute, proper and genuine ho­mogeneity is among plane angles: i. e. no part of a plane angle can be any other then a plane angle, how great, or little soever, and whe­ther proportionable, or improportionable one to another, i. e. whether mathematically ho­mogeneal, or heterogeneal: yet if we seriously consider what is this homogeneity which is among plane angles, that all their parts are plane angles, it is not as is said any quantita­tive, or mathematical homogeneity, the contra­ry of which is plentifully demonstrated in Geo­metry to be possible; nor any such homoge­neity in respect of the manner of their posi­ture and figuration, as to exclude all farther distinguishableness in respect of figuration; but only denotes that in every plane angle, and in every part of every plane angle, the sides lye still in the same plane: which homogeneity, as is plain, excludes neither heterogeneity, in respect of figuration, nor in respect of pro­portion and identity in the way of measuring their quantities.

[Page 166]To the objection that in fig. 12. the recto-convexe angle of Contact BAF can be added to the right right-lined angle BAG, so ma­king the outer angle of a semi Circle FAG; or taken out of it, Sc. the recto-convexe angle of Contact BAD out of the same right right-lined angle BAG, so making the inner angle of a semi Circle GAD: and that therefore the recto-convexe angles of Contact FAB and BAD and the right right-lined angle BAG, and the two angles of the semi Circle GAD, and FAG, are all of them homogeneal and of the same kind; I answer. First what need is there of such endeavours, for you to prove their ho­mogeneity, it being Geometrically demonstra­ted and confessed that there is no analogy, or proportion between them; I mean between the recto-convexe angles of Contact and either the right right lined angle, or either of the angles of the semi Circles? And according to your opinion that which is added or taken out is said to be nothing. But especially its thought strange, why there should be such doubting, that heterogeneals can be added and laid up together as into one repository; it being with as easy connexion performable, as is usual in the addition of incommensurables and specious quantities of which it is not known, whether they be homogeneal, or heterogeneal. And out of an heterogeneal sum, as a store-house, [Page 167] why cannot some of the heterogeneals be sub­ducted, the rest remaining? and what is more usual then the adding of heterogeneal figures one unto another? and subducting out of a given figure some other figure, which is quite heterogeneal to the first given figure? So to adde together numbers, and measures, and weights? the sum of which may be divided, multiplyed, increased, or lessened, notwith­standing its heterogeneity. As supposing A B, C, D, all heterogeneal, as is usual in ana­lyticks, the half, or third part, or any pro­portionable part of this heterogeneal sum may be given: and any one of the heteroge­neal magnitudes subducted, the rest remaining: or a fifth heterogeneal magnitude added to the former sum: or any Algorythme, ever spe­ciously, sometimes compleatly and absolute­ly thereupon performed.

Besides upon geometrical demonstration and your own confession, all recto-convexe, con­vexo-convexe and citradiametral concavo-con­vexe angles of Contact must necessarily by your own principles be allowed to be absolutely heterogeneal to all right-lined angles what­soever; your self acknowledging that neither in equality, nor in any kind of multiplicity, or submultiplicity is any proportionableness possible amongst them. And where between an­gles a mathematical homogeneity is confessed [Page 168] and allowed, yet heterogeneity in respect of their Schematismes and figurations is undenia­ble.

The things therefore constituting and distin­guishing plane angles in respect of their figu­ration are, as above, their sides, their incli­nations, or rather inclineablenesses, and their concurrence. That when two angles have all these in the same respective kinds, the angles are upon good reason in this sense concluded to be homogeneal: but when between two an­gles is an heterogeneity in any of these things, which are of the essence and constitution of an angle, those angles may justly be judged in this sense to be heterogeneal. And that such a specifick heterogeneity may be in each of these, may easily be declared as above. As first in lines which are the containing sides, how easy is it to discover such an heterogeneity? For though a right line, and a crooked-line agree unde­niably in the general nature of a line, and of length, and of extension, yet the rectitude of the one and the curvature of the other, are several kinds of positure, into which the length of the one, and of the other is disposed: that except, in contrarieties, we can see nothing but homogeneity, such an heterogeneity must needs be acknowledged between them. And whereas homogeneity, as to sides, inclination and concurrence, is required to the homoge­neal [Page 169] figuration of angles; the heterogeneity of the sides hinders the possibility of ever making them out to be such; or that by any altering their divarication, keeping their present pro­perties they can be coaptable. And that angles contained by heterogeneal sides may be equal, proves only the equality of the inclinations in either, but not the homogeneity of the figu­ration of the angles, or inclinations; as the equality between a square and triangle in re­spect of their equal perimeters, area's, heights, bases, proves not in the least the homogeneity of their figures. And as right-lines and crook­ed-lines are heterogeneal, as above, not possi­bly to be coapted, with the precedent limita­tions; So also are all curve lines, whose curva­ture is unequal and unlike, nay though it may be they be but several parts of the same line, or though the curvature of both, be every way, and every where equal and like, yet if the convexe and concave parts of the one be not alike posited as in the other, there will be a manifest hete­rogeneity in them, and an impossibility of coapt­ing them, observing the limitations as above.

And why doth the heterogeneity in the sides make heterogeneity in the angles, but because thereby is founded an heterogeneity in the in­clinations, or rather in the inclinablenesses of the one side to the other? For here, it is not the several degrees of the same kind of inclina­tion [Page 170] that is intended; for then all unequal right-lined angles should be altogether hetero­geneal one to another: but it is a more then gradual, a specifick distinctness in their inclina­blenesses, which we are now discovering to make the figuration of the angles more fairly and fully heterogeneal. And as inclination is the habitude of line to line, not being posited in the same right line, nor parallel, for even perpendiculars are in this sense here said to be inclined; so, as above, from the heterogeneity of the lines will arise an heterogeneity of incli­nations: and indeed for no other reasons do he­terogeneal lines make heterogeneal angles, but because their inclinations are necessarily hetero­geneal. And, as above, heterogeneal inclina­tions being respectively equal, as in some right lined and crooked lined angles, this doth not in the least annul the heterogeneity of their in­clinations; as a right line and a crooked line may be equal, yet as to the positure of their ex­tension they are heterogeneal.

And as heterogeneity of sides, or inclinations makes heterogeneity of angles, so likewise doth any heterogenealnes in the other point re­quisite to the nature of an angle; which is the manner of the sides concurrence. And there are only three wayes in the concurrence of the si­des of angles, according to which they can be heterogeneal one to another. For either the [Page 171] production of the one concurring side becomes coincident with the other concurring side, or else it departs from it on the same side on which it did occurre; or else it departs from it on the contrary side to that on which it did occurre: all which are clearly not several degrees of the same manner of concurrence, but several kinds of concurrence: Viz. the one by way of Con­tact, the other by way of Section, and a third by way of curvature, or coincidence. That as these are diversified in angles, I mean from kind to kind, not from degree to degree, so there is thereby lodged in their figurations an heterogeneity, though in some mathematical respects, neither sides, nor inclinations can sometimes be denyed to be however homoge­neal. So particularly angles of Contact in re­spect of their figuration must necessarily be acknowledged clear of another kind, then all other angles: because the inclination of their sides is tangent, concurring only in a punctual touch, whereas the inclination of the sides of all other angles is secant, and at the point of their concurrence by reason of their inclination they cut one another, or else they are coinci­dent; then which, what can make a more material difference in the inclination of the sides? And as more especially relating to that so much urged analogy between right-lined angles and angles of Contact; the inclination [Page 172] of the conteining sides in every angle of Con­tact is such as is impossible to be between the sides of any right-lined angle: for the sides of no right-lined angle can touch without cut­ting. And what more manifest and material difference can be in the inclinations made upon, or unto a right-line, then if in the one case a right-line be inclined unto it, and in the other a circumference, or other crooked-line? Yet further to clear that differency of kind, which is between angles of Contact and other angles; I think on all sides it will be judged unreasonable to make those angles of the same kind; which have neither one common way of measuring, nor are coaptable, nor any way proportionable one to another, nor can any way by the contraction, or dilatation of their sides be made equal one to another; and this we shall find to be the condition of many angles one in relation to another. However mis-under­stand me not, as if I made any commensurabi­lity a full mark of a full homogeneity: for as before crooked-lined and right-lined angles may be equal and of different kinds; having their inclinations different in the kinds of their figurations, though equal in the recesses of the sides.

And thus having at large deduced the grand difference which is between the mathematical heterogeneity of angles, and their heteroge­neity [Page 173] in respect of their figurations, it will now be easy for us to extricate our selves out of all the difficulties with which former Disquisitions upon this subject have been involved.

As first what is to be understood by the equa­lity, which is asserted to be between right-lined, and isoclitical concavo-convexe angles. For it is out of controversy and on all hands yielded, that to any right-lined angle given may be given also a concavo-convexe isocliti­cal angle equal; and that also, in a thousand varieties; as is most manifest, in the circum­ferences of any two and two equal Circles, or any two and two equal Arches. And so in a converse manner; to any isoclitical concavo-convexe angle given, whose sides make their recesse one from the other by an Arch less then a semi Circle, may be given an equal right-lined angle: although in the infinite number of right lined angles, it is impossible to find any more then one right-lined angle equal to the given concavo-convexe angle; because in recti­tude there can be no diversity, as there may and is in curvatures. Now in the above recited cases, why is equality between such different angles asserted possible? and what is meant by their equality? and whence, and how is the equality of them to be demonstrated? Of ne­cessity it must be founded upon some special method of measuring angles, or of somewhat [Page 174] which is in some, if not in all angles; of which in common both these different sorts of angles are naturally and indifferently capable. And to be short, particular and plain; all the myste­riousness of this their equality is founded upon this: that these two sorts of angles, right lined angles, and concavo-convexe angles of equal arches, they both have in common one spe­cial property, of which all other sorts of plane angles whatsoever are destitute. Viz. that each in their kind are isoclitical angles, and the sides in each are isoclitical, and in each angle the one side by the adduction, contra­ction, and drawing together of the sides will be coincident and coapted unto the other. And as the coincidence of right lines the one upon the other makes a right lined angle of Con­tact impossible, so the coincidence of isocliti­cal crooked-lines the one upon the other makes an isoclitical angle of Contact impossible, ex­cept only in an ultradiametral positure. And as the mensuration of right lined angles is by the Arches of Circles drawn upon the angular point intercepted between the two isoclitical sides, to shew how far they are departed from their coincidence; so in isoclitical crooked lined angles, by the same way of mensuration an ac­count may be taken of the departure, which each isoclitical side hath made from the other since their coincidence: and this is the point in [Page 175] which their equality consists and is accounted, and which founds the mathematical homo­geneity which is between them. To instance in the case which is most manifest; in fig. 18. from the angular point A, let the two arches ABC. and AFH of equall circles constitute and contain the isoclitical concavo-convexe angle CAF, and let the arches ABC and AFH be equall: then thorow the points C and H draw the two right-lines AD and AG. According to what is above delivered; it is on all hands agreed, that the isoclitical concavo-convexe angle CAH is equal to the isoclitical right-lined angle CAH: as is copiously demon­strable from the equall arches of Circles drawn upon the angular point as center, cutting all the four lines: viz. the Arches comprized be­tween the two isoclitical crooked-lines, are still equall to the respective arches comprized between the two isoclitical right-lines. For example in the chord AH take any where at pleasure the point I and from the center A draw the arch IB cutting the arch AFH in the point F and the right-line AEC in the point E and the arch ABC in the point B. The arch BF be­tween the two isoclitical arches ABC and AFH is still equall to the arch EI intercepted between the two right-lines DA and GA. For the arches ABC and AFH being equall in equall circles, the right-lines AC and AH are equall: and al­so [Page 176] AE semidiameter is equall to AI semidia­meter: and by the converse of the same ratio­cination AB arch is equall to AF arch: so as in short by superposition, or adaptation the arch BE will appear to be equal to the arch FI: and therefore adding the common arch EF; the arch BF intercepted between the two iso­clitical crooked-lines ABC and AFH is equall to the respective arch EI intercepted between the two isoclitical right-lines: and this where­soever the point I be taken in the right-line AH. So as by this common way of mensura­tion, common to both these sorts of angles, by reason of the isocliticalness, and the coaptabi­lity and coincidibleness of the sides in each, the one being an isoclitical concavo convexe angle is copiously demonstrated to be equal to the o­ther being a right-lined angle. But now after what manner are we to understand this equality asserted between such right-lined and isocliti­cal concavo-convexe angles? It is not an every way absolute equality which is between the angles, such as is between two equall squares, or two like and equall triangles, or any two regular and equall figures of the same kind or, to come nearer to the matter, it is not such as is between two equall right-lined angles, or between two equall, isoclitical, concavo-con­vexe angles, all whose four sides are all of them isoclitical each in respect of all the rest: [Page 177] but as things that are like each other, are like only in some things, and unlike it may be in many others; such is the equality between any two such angles; Viz. only a respective equa­lity, such as is possible among heterogeneals, and inferring a necessity of some other respe­ctive inequalities. And such an equality may be between two mere heterogeneals; they may be of equal length, and different breadth, or weight: so a Triangle, and a square and a Circle may be all equal, either in perimetry, or surface, but not in both; so a right-lined angle, and an isoclitical concavo-convexe angle may be equal in respect of the recesses which the isoclitical sides make each from other and from their coincidence and coaptation, but in other respects they want not their manifest inequalities and heterogenealness. As a solid to a solid may have equal proportion that a line to a line, yet solids and lines are heterogeneal: so a crooked-line from a crooked-line may make equal recesses, as a right-line from a right-line, and yet in many other things much heterogenealness may be in the angles which they constitute. You will say wherein? I answer in the rectitude and curvature of the containing sides. And in these different respects two iso­clitical concavo-convexe angles may be both equal and unequal the one unto the other, Viz. equal in the recesses of the sides, but une­qual [Page 178] in the curvatures of the sides: in the same manner as two figures may be equal in their perimetry, or superficial, or solid content, and yet be figures of different kinds under diverse inequalities: as the one a Rhom­bus, the other a square, the one a Cylinder, the other a Dode [...]aedron. So a thousand con­cavo-convexe isoclitical angles may be equal in respect of the recesses of the sides, yet each of a several kind: as a thousand figures different in kind may be equal in perimetry, height, base, superficial, or solid content. But you'll say, what is the rectitude, or curvature of the containing sides to the nature of angularity? I answer, they are of essential concern to the limiting and determining the nature of an­gles: angularity being the habitude of concur­ring lines each in respect of the other, as to their concurrence and inclination. And though the inclination of isoclitical crooked lines may be equal to the inclination of right-lines one upon another, in respect of the equal recesses and departures which the isoclitical lines make each from the other; yet there still remains a vast inequality, dissimilitude, and unanalo­gableness between the angles and their incli­nations, in respect of that little of figuration, without which neither can an angle be consti­tuted, nor an inclination made: in a word the sides may make equal recesses, yet be unequal [Page 179] in their curvature, and unlike in their figura­tion: and neither by imagination, nor cir­cumduction, nor any other operation can the one possibly be reduced, or coapted to the other, without setting the homologal points at improper and undue distances and positures one from another; which shews a specifical difference between the two inclinablenesses of the one and the other: besides that a right-lined angle can continue its inclination between the sides infinitely, but many isoclitical con­cavo-convexe angles thereunto equal by the ne­cessity of their curvature must terminate within a very little space: circumferences and several other arches, not being possible to be produ­ced beyond their integrity; so as some three given angles constituting a given triangle as to its angles, cannot, in like manner, con­stitute a triangle of any given magnitude; which is otherwise in right-lined angles. And that the equality between isoclitical concavo-convexe and right-lined angles is not so absolute as to make them every way alike, equal, and of the same kind, may appear especially in this, which is elsewhere demonstrated; Viz. that an ultradiametral concavo-convexe angle of Con­tact, being isoclitical, is alwayes equal to two right right-lined angles, which no one right lined-angle can be: and if it be anisoclitical of the larger, it is ever greater then two right [Page 180] right-lined angles can be, which is impossible also for one right-lined angle to be.

And the difference between mathematical heterogeneity and the heterogeneity of angular figurations being as above discovered; the na­ture as well of anisoclitical crooked-lined and mixt-lined secant angles will as clearly ap­pear: Viz. that comparing them with right-lined angles they are compound and concrete angles constituted of right-lined angles and angles of Contact; which are demonstrated every way heterogeneal: and such anisocliti­cal secant angles cannot be divided into any number at pleasure of parts homogeneal either mathematically, or in respect of their figura­tion, but of necessity some of them must be both wayes heterogeneal. This is manifest, because the fluxe, or circumduction of angles of Contact or of one of their conteining sides addes only a right-lined angle to them: after the same manner as the four right right-lined angles, which compleat the space in any plane, about any given point, may be exhausted by the circumduction of a crooked, concave, or convexe autoclitical, or antanaclitical line, as well as by the circumduction of a right-line. And that this ties not both angles to be of the same kind, may easily appear from the hete­rogeneity between lines and their fluxes, which are superficial; or surfaces and their fluxes, [Page 181] which are solid. No wonder therefore if by the fluxe of an angle of contact, or of one of its sides, be created another kind of angle, holding no analogy with the former: the he­terogeneity and improportionableness of right-lined angles and angles of Contact having been demonstrated.

So upon the same ground we may be assisted to look into the special properties, conside­rable in the several kinds of angles peculiar unto some and incommunicable unto others, for example.

In right lined angles, neither the greatest possible angle, nor the least possible angle can be given; though all usually said to be within the compasse but of one kind. But to pursue the difference which is in angles; angles of Contact, except such as are contained under lines of the same rectitude and curvature are every one of a several kind, either mathema­tically, or in respect of their figuration, or both. And except convexo-convexe angles of Contact of equal arches, which may be di­vided into two equal angles by common right-lined tangents; all other angles of Contact are, every one, both the greatest, and least possible, of their special kind: and every angle of Contact contained under the convexe side of its arch, or arches, is the least possible under those sides; which I suppose was the spe­culation [Page 182] unhappily missed by those learned men, who would have imposed upon the world, upon that their mistake, the dream of the coincidence of the sides in such angles.

Again a right-line may be drawn dividing a convexo-convexe angle of Contact, whether it divide it equally, or unequally, but a right-line cannot be drawn dividing either of the recto-convexe angles of Contact into which the former was so divided; whether the two recto-convexe angles of Contact be of the same, or different kinds: as is up and down demonstra­ted in Geometry.

In citradiametral concavo-convexe angles of Contact, either the Arches are of unequal cur­vature, or which is tantamount, though they be of equal curvature, yet they touch not at homologal and answering points, being not all over of equal curvature; which makes them notwithstanding the respective but not answe­ring equality of their curvature to be anisocli­tical. And between their Arches containing the angle of Contact, a right-line cannot be drawn; but infinite crooked-lines in number may be drawn, bearing in like manner their convexity towards the concave which is in­ward, and their concavity towards the con­vexe of the other side, which is also inward.

Concavo-convexe angles concurring by way of section, and having two right-line tangents [Page 183] drawn upon the Arches at the angular point are equal unto the right-lined angles contained under those right-lined tangents; adding re­spectively to each right-lined angle one of the recto-convexe angles of Contact, and subdu­cting out of it the other recto-convexe angle of Contact: and when those two recto-convexe angles of Contact are equal, as they are, when the sides are isoclitical, then the concavo-con­vexe angle is exactly equal to the right-lined angle: but when the two recto-convexe angles of Contact are unequal, as they are when the sides of the concavo-convexe angle are anisocli­tical then the concavo-convexe angle and the right-lined angle are unequal.

Concavo-concave angles concurring by way of section (as all such ever do; or by coinci­dence, and then one right-lined tangent gives the analysme of them, shewing the two recto-convexe angles of Contact, by which the an­gle of coincidence is less then two right right-lined angles) are by two right-lined tangents at the angular point reduced into the right-lined angle; which is the least of those right-lined angles that are greater then it, exceeding it only by two recto-convexe angles of Contact to be taken out of it.

Convexo-convexe angles concurring by way of section are by two right-lined tangents at the angular point reduced into a right-lined [Page 184] angle; unto which to make it equal to the con­vexo-convexe angle, are to be added two recto-convexe angles of Contact. And the right-lined angle is the greatest of all the right-lined angles that are less then the convexo-convexe angle.

Recto-concave angles concurring by way of Contact are the greatest angles possible under those two sides.

Recto-convexe angles concurring by way of Contact are the least angles possible under those two sides.

If we compare recto-concave angles of Con­tact with right lined angles, they are less then two right right-lined angles by one only recto-convexe angle of Contact. And the least recto-concave angle of Contact is greater then the greatest right-lined angle whatsoever.

Recto-concave angles concurring by way of section compared with right-lined angles which are constituted, i. e. compleated by the right-lined tangents drawn upon the Arches at the angular points; are less than such respective right-lined angles by a recto-convexe angle of Contact.

Recto-convexe angles concurring by way of section compared with right-lined angles which are constituted, i. e. compleated by right-line tangents drawn upon the Arches at the angular points; are greater then such right-lined [Page 185] angles by a recto-convexe angle of Con­tact.

Every angle of curvature, or coincidence, having a right-line tangent drawn upon the angular point, appears to be less then two right right-lined angles by two recto-convexe angles of Contact.

The inclination of the sides without the an­gular point at any two respective, or other points, the one taken in the one side, the other in the other, is very nearly shewn, and as nearly as is possible in right-lines, by the right-line tangents of those respective points: but in mixed lined and mixed crooked-lined angles by several wayes of accounting, several points are made to answer one another, as by accounting by distance from the angular point, or by accounting by equalness of lines along the sides &c.

Mixed lined angles of Contact, when they can be, and are, divided by a right-line, the parts are heterogeneal and unequal: and one of the unequal parts is a right-lined angle.

Every recto-convexe, and convexo-con­vexe, or citradiametral concavo-convexe an­gle of Contact is the least possible under those sides.

Rectilineary mensurableness in mixt lined, crooked-lined angles concurring by way of section, begins from the recto-convexe angle [Page 186] of Contact: as in right-lined angles from coin­cidence.

A convexo-convexe angle of Contact, in re­spect of dividableness by right-lines is an angle made up only of heterogeneal parts, when it is a mixt-crooked-lined angle: but when it is an unmixt crooked-lined angle, it hath some parts which are homogeneal, Viz. two equal recto-convexe angles of contact, which are therein added the one unto the other. And those two equal recto-convexe an­gles of contact, as they are homogeneal, I mean of the same kind one with another, both mathematically and in respect of their fi­guration; so mathematically they are homo­geneal and of the same kind with the convexo-convexe angle which was divided; but in re­spect of it, as to their figuration, they are he­terogeneal and of another kind.

The most simple angle may be divided into heterogeneal parts: i. e. the inclinableness of the one side to the dividing line both in respect of figuration and proportion may be specifically different from the inclinableness of the other side to the same dividing line: as a pentagone may be divided into a tetragone and a triangle, so a recto-convexe angle of Contact may be di­vided into two parts heterogeneal the one to the other, and to the first angle of Contact, both in respect of figuration and proportion: [Page 187] viz. into a new recto-convexe angle of Con­tact, and a concavo-convexe angle of Contact. Therefore no angle can be said homogeneal in that sense, as if it could not be divided into parts heterogeneal; whether you please to un­derstand it, in respect of mathematical homo­geneity, or positure, and figuration, or what respect soever else that limits and distinguisheth plane angles one from another.

And to give a brief and general account of the comparative admensuration of angles, as not being right-lined, yet by way of compara­tive admeasurement, they may in respect of their rectilineary parts be reduced and referred to those that are right-lined; the containing sides not being right-lines, at the angular point draw right-line tangents touching the arch, or arches in the angular point; and the right-lined angle contained by those right-lined tangents will be, as to the recesses of the sides at the an­gular point, either equall unto the first propo­sed angle, or the least right-lined angle greater then it, or the greatest right-lined angle lesser then it: or if two right-lined tangents cannot be thus placed at the angular point, either the first proposed angle was a mixt-lined angle of contact, said, if a recto convexe to be less, if a recto-concave to be greater then any right-lined angle; or else it is a crooked-lined angle of contact, which if convexo-convexe, or con­cavo-convexe [Page 188] and citradiametral is less then any right-lined angle, but if concavo-convexe and ultradiametral, is greater then any right-lined angle, nay sometimes equall to, or grea­ter then two right right-lined angles: or else it is an angle of coincidence, or curvature. All which is to be understood to shew the inclina­tion of the sides at the angular point, as the chief for use in Geometry, but not necessarily else-where. So crooked-lined, or mixed lined angles are compared with right-lined angles by drawing at the angular point right-lines touch­ing the Arches there, and comparing the crooked or mixed lined angle with the right-lined angle so constituted, respectively adding, or subducting the recto-convexe angles of Con­tact hereby created: and this, whether the Arches be isoclitical, or anisoclitical, or how­ever posited. So all crooked, or mixed-lined angles concurring by way of section may have a right-lined angle given, which if it fall short of equality is either the least of the right-lined angles that are greater, or the greatest of the right-lined angles that are less then the first crooked-lined, or mixed lined angle. And so an analysme may be made of the greater angle into its heterogeneal parts; and the crooked lined, or mixed-lined angle may be reduced unto, or compared with right-lined angles, only with the addition, or substraction of recto-convexe [Page 189] angles of Contact, being angles less than the least right-lined angle whatsoever.

All angles have their inclinations compoun­ded of the inclinations of the interjacent lines each to other in order: and of the inclinations of the sides to the lines next adjacent to them; which composite inclination may be heteroge­neal, as well as homogeneal in respect of the inclinations of which it is, or may be com­pounded.

Equally arched convexo-convexe, or con­cavo-concave angles, may by a right-line be divided into equal parts mathematically homo­geneal, but heterogeneal in respect of their figuration: but such angles cannot be divided into any more, or any other equal parts, for the reason immediately to be subjoyned.

Heterogeneals taken together in several con­cretes proportionably, Sc. each respectively in the same proportion, they hold exact pro­portion, concrete to concrete: as double cube and double line, are double, to single cube and single line, Viz. the concrete to the con­crete: but set them out of the same respective proportions and the concretes are no way pro­portionable, or in analogy, concretely to be compared: as double cube and treble line, are in no proportion, to single cube, and single line. So double number, and double weight, and double measure, the whole concrete, is [Page 190] double, to single number, single weight and single measure: but setting them out of the same respective proportions; double number, and double weight and treble measure, being all­together concretely taken, are mathematically heterogeneal and improportionable to single number, single weight and single measure; being in like manner concretely taken: because the heterogeneals in the one concrete [...]old not the same respective proportions to the answe­ring heterogeneals in the other. So convexo-convexe, or concavo-concave equally arched angles being secant, hold proportion when divided equally, as they may, by right-lines: but they are merely heterogeneal and without proportion, when divided by a right-line une­qually. The ground of which is the hetero­geneity of the parts, of which such concrete angles are made up when compared with an­gularity constituted by right-lines; which he­terogeneal parts, when the angle is divided equally in two by a right-line, are in the con­cretes, each respectively in the same propor­tion; so making the concretes, though of he­terogeneal parts, to be mathematically homo­geneal and proportionable one unto the other: but when the angle is unequally divided; in the two concretes the heterogeneal parts, of which they are made up, are not respectively in the same proportion; for the recto-convexe [Page 191] angles of Contact, in the concrete angle, are divided equally, and the right-lined angle, which is in the concrete angle is divided une­qually: so making the parts of the divided an­gle mathematically heterogeneal and impropor­tionable; because the compounding hetero­geneal parts are not respectively according to the same proportion divided.

The least possible angle under any two given incoincidible lines is the least angle of Con­tact which is possible under them.

Recto-convexe, and citradiametral conca­vo-convexe angles of Contact as they are the least angles possible under their sides; so they are indivisible into parts holding all each to other any thing of mathematical homogeneity, or proportionableness: Like unites they may be multiplyed to any proportion as whole num­bers, and separately set: but cannot at pleasure be divided, nor at all into parts which are all of them mathematically homogeneal: nor can any number of them be at pleasure adjoyned one to another. So to any right-lined angle another may be imagined in any proportion, but sometimes it must be the composition of se­several angles, and more then can stand at the space circumjacent about any one point. It not being possible for above four right right-lined angles to stand about the same point. As to a given point, other points may be given in [Page 192] any proportion, as whole numbers; but they must not then be adjoyned one to another: but properly by the dividing of a point, farther proportions are not to be expected. So in such angles, though heterogeneal divisions be infi­nitely possible; homogeneal, I mean mathe­matically, being impossible, are not, as above, to be expected.

Two unequal anisoclitical angles contained under the same two anisoclitical sides, though in respect of their figuration, both are formed upon the same inclinableness of the sides one to another, yet they are mathematically hete­rogeneal: because being two concrete angles made up of heterogeneals, viz. of the angle, or angles of Contact and right-lined angles; in the two concrete angles, the angles of Con­tact are in the proportion of equality, and the right-lined angles in the proportion of inequa­lity: so as the concrete angles can have no proportion the one to the other.

The half of the inclinableness of any arch upon its self, i. e. upon another arch like and equal, is still comprehended and contained under the inclinableness of a right-line upon the same arch: that though their figurations are ever heterogeneal, and their inclinations can never be equal, yet they may be mathematical­ly homogeneal: as different numbers are al­wayes unequal though ever proportionable.

[Page 193]To no anisoclitical angle, whether of Con­tact, or concurring by way of section can any right-lined angle be made equal. And generally between isoclitical and anisoclitical angles equality is impossible. And when anisoclitical angles are compared with those that are iso­clitical as greater, or lesser; it is not to denote in both any mathematical homogeneity, and that so by possibility they are reducible to a true, compleat and analogous equality; but the intent is only to declare, whether of them hath, or can have the containing side, or sides falling within, or without the other; which is only from their inequality by the whole kind. So though anisoclitical angles may be greater, or less then a right-lined angle, yet between the one and the other there is neither common way of measuring, nor any proportion, though both of them be quanti­tative plane angles. There is no common way of measuring them, because of the anisocliti­calness, and if there could be any proportion between them, then might a right-lined angle be given equal to such anisoclitical angles; the contrary of which is otherwise clearly de­monstrable. So as a thousand angles quantita­tive by conf [...]ssion and having right-lined an­gles lesser and greater then themselves, yet can have no right-lined angle equal to them.

A right-lined angle cannot have its inclina­tion [Page 194] at the angular point, much less all along the sides divided equally by a crooked-line, whether autoclitical, or antanaclitical, or of what curvature soever: though any such crooked-line may from any angle divide equal­ly the plane bounded within a right-lined triangle: for if in fig. 17. BAC be a right-lined angle: I say, no crooked-line &c. can divide it at the angular point equally. If it be possible let it be divided into equal parts by the arch AGE whether autoclitical, as when it is conceived to be part of the arch AGEF, or antanaclitical, as when it is conceived to be part of the arch AGEHD. Then let the right-line AD divide equally the right-lined angle BAC. It is manifest, if the crooked-line AG never occurre with the right-line AD, that then it doth not divide the angle BAC equally; for the right and crooked-lines cannot be coapted. Let therefore, if they do occurre, the place of their first meeting, or occurrence be at E. Therefore the arch AGE falleth within the angle BAD and therefore divides the whole angle BAC unequally, whether it be autoclitical, or antanaclitical. And by the same demonstration appears that it is as impossible for it to divide the inclination all along the sides equally. And though a crooked-line may divide any right-lined figure, because of its perfect bounds, from any angle [Page 195] into equal parts; yet this no way evinceth any possibility of dividing a right-lined angle by a crooked line into equal parts: and though from any point of such a dividing crooked line, lines might be drawn making up a figure, whose parts divided by the crooked line from the angle are equal; yet this is no more then is performable upon any point of any crooked line drawn between the sides of a right-lined an­gle at randome, whether the parts of the an­gle be homogeneal, or heterogeneal, equal, or unequal.

Difference of curvature by a perpetual ne­cessity infers difference of inclination, whether the curve line be inclined upon a right-line, or upon a curve line; for still the one of the curve lines will fall within, or without the other. So let inclination upon a right-line be in a recto-concave angle of Contact, recto-convexe angle of Contact, recto-concave, or recto-convexe secant angles; by no right-line can any of those inclinations be made upon the first right-line, nor by any other crooked-line: but still the lines will fall either within, or without▪ And all angles of Contact under lines of different curvature and rectitude, as they are of [...] thousand Mathematically and extra-quantitatively different kinds one from another, so they are [...] manifestly distinct in kind from all right-lined angl [...] whatsoever. [Page 196] And as every number is a different kind, in like manner as to positure and figuration is every line a distinct kind differing from all other not agreeing with it in rectitude and curvature; whether the curvatures be homogeneal, i. e. every where equal, as in Circles, or hetero­geneal, i. e. unequal in the several parts, as in ellipses, hyperbolas, parabolas, &c. And accordingly judgment is to be made of the sides under which angles are contain'd, and the an­gles contain'd under them: alter the kind of either side, and the kind of the angle is chang'd; ever, in respect of figuration; and most what, mathematically; because the former inclinablenes of the sides is taken away, and a new kind of inclinablenes introduced between them. Hence appears; no two recto-concaves, no two recto-convexes, no two concavo-convexes, no two convexo-convexes, being all angles of Contact can be equal, except their sides have the very same rectitude and curva­ture. In general, equality is not to be asserted between angles, except either for the men­surablenes of both in some common way of measuring, or at least because they can be so cast into a coaptation, as that either shall con­tain all the quantity which is in the other. And as he erred in squaring the superficial con­tent of a circle, that in his quadrature left out a small lunular figure; so neither can he be [Page 197] justifyed to have given one anisoclitical angle equal to another, who, as he must of necessi­ty, takes in, or leaves out an angle, or some angles of contact which are in the one and not in the other.

Particularly no mixed lined, nor mixed crooked lined angles whatsoever, whether secant, or of contact, can by any possibility, either by right-line, or crooked line be divided into equal parts, or angles, whether of the same, or of different, kinds. And in few, those angles, which cannot be divided into two equal parts cannot be divided into three, fower, five, or any other number of equal parts.

The comparative admensurement, as above, of right-lined and not right-lined angles is as well of their heterogeneal inequality, when he­terogeneal, as of their homogeneal equality and proportionablenes when homogeneal.

All not right-lined angles comparatively admeasured, to right-lined angles, only with the addition, or only with the subduction of any angle, or angles of contact, are hetero­geneal to all right-lined angles, as also, if when one angle of contact is to be added, and another subducted, the two angles of contact be unequal. And as is manifest, of several not right-lined angles, that is still the greater, whose comparative and relative admeasure­ment [Page 198] is made to the greater right-lined angle: and of those, whose comparative admeasurement is unto equal right lined angles, they are grea­ter, or equal, or less according to the equality, or inequality, and comparing together of their angles of Contact▪ And of concavo-concave angles the right-lined angle cannot be given, which is the greatest of those right-lined an­gles that are less then the concavo-concave angle: nor in convexo-convexe angles can be given a right-lined angle, which is the least of those right-lined angles which are greater then the convexo-convexe angle: nor in recto-concaves the greatest of the less: nor in recto-convexes the least of the greater: whereas in concavo-convexe angles, the right-lined angle to which the comparative admeasurement is made, may according to the case be either equal, as when the sides are isoclitical; or the greatest right-lined angle that is less, as when the concave side is of less curvature; or the least of the greater, as when the concave side is of greater curvature.

From these things may appear how that objection is to be answered, in which you urge that there is a proportion between mixed lined angles, whose sides are seca [...], and right-lined angles, because the one by its multiple may exceed the other; and that therefore an homogeneity, mathematically to be under­stood, [Page 199] is to be acknowledged amongst them. To this is answered that all anisoclitical angles whatsoever having their sides concurring by way of section when compared unto right-lined angles are heterogeneal compositions of angularity, i. e. cannot be divided into any number at pleasure of parts all equal; but as is manifest, they in that their relative nature are concrete and composite angles, formed of right-lined angles by adding to them, or sub­ducting from them mixed lined angles of Con­tact: so as the right-lined part of the secant anisoclitical angle, or any the least part of it hath true proportion with all right-lined angles; and the whole being heterogeneal, yet hath a semblance of proportion with right-lined angles, because of those parts which it hath which are homogeneal with them; but between the other heterogeneal part which is the mixt-lined angle of Contact, and right-lined angles, neither is, nor can be any proportion: and it is because of this part that it is not a true, but a seeming proportion only, which is be­tween the whole concrete heterogeneal, com­posite anisoclitical angle, and right-lined an­gles: for if it were a true proportion and pro­portionablenes which were between them, it should then be possible to give a right-lined angle equal to such an anisoclitical secant angle; which when done, I have no more to [Page 200] say, being well assured I am able to demon­strate the contrary upon whatsoever right-lined angle shall be offered under that notion. And whereas it is urged that a right-lined angle is, and may be equal to a mixed lined angle; all their difference being only in this, that the sides of the one are more spread and divari­cated then the other; what is this else but to say, that they are equal, saying that the one is bigger then the other. For angles of Contact can divide angularity, or space equal­ly, or unequally into more, or fewer parts: and they are not indivisible, because every one contains innumerably more in it: and by the definition of an angle, lines have a sufficient inclination to constitute an angle, if in the same plane, they lye not both in the same right-line. And to all other cases and objections of the like nature grounded upon them athema­tical heterogeneity of the parts, of which such special angles are constituted in respect of right-lined angles, like answers may be addressed.

Though a right-lined angle cannot be di­vided into equal parts of the same kind mathe­matically, or in shape by a crooked-line, nor a mixed, or crooked-line angle by a right-line into parts of the same kind in respect of figuration; yet some crooked lined angles may be divided into parts exactly equal of the same kind one with another, and with the whole [Page 201] mathematically by a right-line. For it is ap­parent, there may be between magnitudes a sufficient homogeneity for proportionablenes without excluding all further discriminableness between them: as between equal arched con­vexo-convexes, or concavo-concaves and their parts when they are equally divided. So be­tween lines of all kinds, and numbers of all kinds, is proportionality. Yet still their propor­tionality is in respect of somewhat which is homogeneal in them. Viz. that they are all re­soluble into parts that are homogeneal, or parts into which the rest are homogeneally resoluble, or after the same manner mensura­ble, as equal arched convexo-convexe angles into two equal recto-convexes: yet notwith­standing the proportionablenes between the convexo-convexes and the recto-convexes; such an heterogeneity is in their figuration, that by no divarication of sides can they ever be made equal. And though the equal arched convexo-convexes and concavo concaves may be divi­ded by a right-line into two equal parts; by no lines whatsoever can they be divided into any more equal parts nor their recto-convexes by any line whatsoever into two equal parts; such parts of such angles being more impos­sible to be given in Geometry, then the square roots of unsquare numbers, or cubicke roots of uncubical numbers are in Arithmeticke. As [Page 202] any two points, or any two indivisibles may have some few proportions amongst them, but no more: the latitude of proportionablenes being limited by the special natures and kinds of things: so between special kinds of numbers cannot be all proportions: nor between special lines all angles, or all proportions in all an­gles.

Hence also we may clear the possibility of that speculation of the learned persons by you named, that in heterogeneals there may be a passing from greater to less, and from less to greater in a continuous manner without, ever passing thorow equality; however you are pleased to bestow upon it a contemptuous smile, if not an hiss: Viz. that we should assert that mixed lined, or crooked lined angles can pass by the divarication of the same sides from being greater to be less then given angles, or contrarily by their contraction, and yet in that transit, never be equal. What is more obvious then to give instances of anisoclitical angles, less then a given isoclitical angle, and by di­varicating and distending the sides of the ani­soclitical angle, it may be made to exceed the first given isoclitical angle; yet in all the way they could never be equal the one to the other; the one being isoclitical, the other anisocli­tical: coaptation and the passing of the sides from the angular point every where prove [Page 203] their inequality: and let those that assert their equality any way shew and admeasure it. And the sober understanding of those sayings, that such angles passe from less to greater without ever being equal, is not to assert any homo­geneity, or proportion of any kind between them, or common way of measuring their quan­tity in and according to the same indefinite quantity, but only to shew how the sides may pass within and without each other; but be­cause of their anisocliticalnes and difference in respect of rectitude and curvatures, they can never be brought to be coincident. And why should this appear so monstrous, that transi­tions should be, in the sense abovesaid, made from greater to less, without passing thorow equality? seeing it is most manifest that though a crooked-line circumducted about the angular point of a right-lined angle makes infinite divisions of the right-lined angle; yet they are ever unequal, and never by possibility can be equal: though notwithstanding, the same crooked line may divide a crooked lined iso­clitical angle equal to the given right-lined angle into equal parts; which besides plenti­fully shews the heterogeneity which is be­tween equal right-lined and crooked-lined iso­clitical angles, notwithstanding their equa­lity: yet by making up the right-lined angle into a compleately bounded figure, the croo­ked-line [Page 204] may divide the plane of it into equal isoepipedal and isorrhopical parts: as is more easy to demonstrate then that there should be any need to set it down. And it is not to be stranged at, that we assert a crooked-line can divide a right-lined triangle into equal parts, but not any of its right-lined angles: for the figuration of the one is compleat, and the production of the crooked-line, as well as of the lines containing the right-lined angle, in the triangle are limited; all which are quite otherwise in mere angles, being in many kinds of angles and inclinations very alterable by the production of the lines. And if you will pertinaciously say that a crooked-line may di­vide equally a right-lined angle, shew their equality, and your way of admeasuring the equality of the parts: both coaptation of sides, in whose habitude the nature of angles chiefly consists, and also the way of measuring by interjected arches demonstrating and declaring the contrary: Nay sometimes in quantities among which there is true proportion, the special differences of their kinds may be the authour of little less. So commensurable quan­tities, being in their commensurability infi­nitely divisible, may be continually increased, or lessened by quantities vastly, less then any given quantity; yet as they pass from less to greater never can hit equality with any the like [Page 205] quantities of the same general kind, being incommensurable. So in the present question, heterogeneity of sides makes an impossibility to coapt, though by straitning and divarica­ting they may fall within, or without, and so be less, or greater, but being impossible to coapt, and incapable of any other common way of measuring, equality cannot be concei­ved in them; in which is praesupposed a com­mon way of measuring to declare and prove their asserted proportion and equality. And in this whole matter what is more said then that the cutting inclination of a right-line upon a right-line, and of a right-line upon a crooked, or of a crooked-line upon two crooked-lines being all of different curvature, may any of them be greater, or less then one another he­terogeneally, but can never be equal; as to those that will consider, is most clearly mani­fest and necessary. And this is no more strange, then that odde numbers may be less, or great­er then any even number, at least above two, yet by their constitutive nature, they can never be equal: and for the same reason magnitudes commensurable and incommensurable may be greater, and less then one another, and by a less quantity then any quantity that can be given; yet, for the specialty of their natures, they can never be equal, remaining so distinct in their ultimate kind: and however they have [Page 206] many general things in which they agree, yet they are not accountable specially, but by several wayes and algorythmes. And as it is the binding up of numbers to the specifick properties of Evenness, or oddness, and of magnitudes to commensurability, or incom­mensurability, that makes equality between their kinds impossible; so it is the incoapta­blenes and want and impossibility of a common way of measuring, which is between the two and two sides of such heterogeneally unequal angles, keeping all along the properties of their inclination, that renders equality be­tween their kinds impossible; though either may sometimes truly and manifestly be greater or less then the other, however without pro­portion. For in heterogeneals may be a two fold inequality, either an inequality according to proportion, or an inequality without pro­portion: and though they be heterogeneals, if it be a true limited, determinate proportion of inequality which is between them, as be­tween an odde number and an even, a com­mensurable magnitude and an incommensura­ble, they are then considered as some way, or according to something which is, homo­geneal, not heterogeneal, in both: but if their inequality, though real and apparent be with­out any true and homogeneally determinable proportionablenes, as between right-lined and [Page 207] anisoclitical angles, then the consideration had of them is in their heterogeneity. So to the objection, that right-lined angles, and recto-convexe angles of Contact have proportion of greater and less, and are therefore homogeneal, if any quantitativenes be to be asserted in the recto-convexe angles of Contact; and so of the rest: I answer, there is not a proportionable inequality between right-lined angles and recto-convexe angles of Contact, and the like; but an inequality, which is improportionable: as when the Earth is said to be intruth greter then a point; though in many particular hy­potheses contrarily conceived, for the better observing and accounting several phaenomena, and the better accommodating of instruments for the making of observations. So in the ge­nesis of quantities and figures, when one quan­tity hath its genesis by the fluxe, or motion of another, the quantity formed by that motion is greater then the quantity moved, though still without any proportion: so by the diva­ricating of the sides, by the motion of one of the sides, of a recto-convexe angle of Contact, is a genesis of recto-convexe angles of section; and the recto-convexe angles of section are still greater then the recto-convexe angle of Con­tact, or its respectively moved side: and all still without any proportion. And inequality without proportionality being so usually asser­ted, [Page 208] and familiar between mere heterogeneals, doth easily acquit it self from the reproach you charge it withal, of being a contradiction in the very terms; for finite and infinite never were denyed to be unequal, yet never can be made out to be mathematically homogeneal and proportionable. For though all proportion is either of equality, or inequality; that hinders not but an inequality in heterogeneals may be admitted, without an asserting of proportion between them. So as mathematical homoge­neity is not proved by inequality, till propor­tionablenes be as well proved as inequality▪ Of the same leaven is that strange kind of reasoning you use; to any right-lined angle a crooked-lined angle may be made equal (we confess a thousand several crooked-lined angles may be made equal to any one right-lined) angle) but you from thence inferre that there­fore all crooked lined angles and right-lined angles are of the same kind; without adding as well, that to any crooked-lined, or mixed lined angle, a right-lined angle also might be made equal; which can never be done; the contrary thereof being confessedly demonstra­ted in Geometry: or if any be so opinionated that it is either easy, or feasible, let them give a right-lined angle equal to an anisoclitical an­gle, whose sides concurre by way of section or to an ultradiametral convexo-concave angle [Page 209] of Contact: and trye whether the equality of the two angles be not disproveable. In the same manner inequality without proportion, is asserted between both the recto-concave and recto-convexe angles of the semi-circle, i. e. the inner and outer angles of the semi-circle, and a right, or any other right-lined angle▪ But to return, from the concrete and com­posite nature of not right-lined angles, when compared with right-lined angles; the enig­maticalnes of the proposition, that magnitudes of one sort may be greater, and less then a given magnitude of another, but never equal, becomes most clear and doubtles: as infinite solids may be given greater, or lesser then the heterogeneal concrete magnitude of a cube foot and a foot-line; but never can any solid be given thereunto equal: so likewise infinite heterogeneal concrete magnitudes consisting each of solids with the accrescency, or an­nexion of a foot-line, may be given greater, or lesser then a cube elle; but in such compa­risons, never can the concretes of the hetero­geneal magnitudes be equal, or in proportion to any solitary, single, one, of the hetero­geneals.

In like manner it manifestly appears what is to be thought of that often pretended equality between the two angles of a semi-circle and a right right-lined angle: for by coaptation and [Page 210] the goings forth of the lines from the angular point it appears otherwise. And how will they that assert their equality demonstrate it? and by what way will they admeasure it? for in­terjacent and intercepted arches of Circles drawn upon the angular point as center, can in these angles contribute nothing to the ad­measuring of them: and coaptation makes against their equality: and is so farre from ma­king a right right-lined angle to be the con­stant standard of all angles of semi-circles, that it manifestly shews the angles of greater semi-circles to be greater and of less, lesser. The objection, that unequal circles, semi circles, and segments of equal degrees, cannot be judged like and homologal figures, except all their respective angles be equal, as well as the sides homologal: this may be easily answered; Viz. that it is like genesis that makes like figures, which in all right-lined figures makes the answering angles equal, and in all figures the sides to be homologal, and the diffe­rence of the homologal angles to be less then the least right-lined angle.

In Circles the same genesis, by the circum­duction of the semi-diameter about the Center, which makes the figures like, and the sides homologal, makes in unequal circles the cur­vatures, angles of coincidence, the angles of the semi circles, and of homologal segments, [Page 211] necessarily unequal. Or, then figures [...] be judged like when in a concentrick [...] the perimeter, sides and lines of the [...] proportionable to those in the other, [...] or may be placed parallel, or [...] those in the one, to those in the other▪, [...] which in right-lined figures, tis true, th [...] alwayes follows equality between the answe­ring angles, but not so in other lined figures. And because equality of angles in like right-lined figures is so much urged; the difference of the case of angles in like right-lined and like crooked-lined figures may plainly appear in another remarke; Viz. that in like right-lined figures, the sides of the answering angles may from the answering angular point be exactly coapted one to another; which in unequal Circles, though never so like figures, is most apparently impossible: that it cannot but be unreasonable to expect as absolute a conformity between the answering angles in like crooked lined figures, as there is in like right-lined figures; being so manifestly against both the eye and demonstration. And therefore like segments of Circles, are not defined by equa­lity of angles contained between their arches and their chords; but by the equality of the angles contained in them; the one being cer­tain, constant in all, and demonstrable; the other in most cases, not only doubtful, but [Page 212] impossible. So the objection, from the asserted equality of alternate angles, made by a right-line cutting parallel circumferences; is readily answered by denying the truth of what is pre­sumed in the objection, as never by any de­monstrated, or ever possibly demonstrable, Viz. that the alternate and vertically opposite angles, made by a right-line cutting parallel circumferences are equal. And the tyranny of forcing lines out of their natures and special properties, may appear in that very instance of compelling the parallelisme of curve lines to answer the consectaries and idioms of the parallelisme of right-lines: in which, to omit the alleadged instance, as by you unproved, and for good reasons by us to be denyed, a right-line tangent of the lesser concentrick Circle cuts the circumference of the greater, and infinite right-lines cutting the greater, neither cut, nor touch the lesser, which is repugnant to the nature of parallelisme in right lines. That that which is so much contended for, that a crooked-line and a right-line are homogeneal as to length and their general lineariness, was, or ought never to be denied, there being all possibility of equality, and truly proportionable inequality between them, what kind of curvature soever the crooked-lines bear: but that they are homogeneal as to the positure of their longitude; the site and [Page 213] manner of their extension, hath unto me been alwayes unconceivable: whence the truths on both hands clearly follow, Viz. that an arch and a right-line may be equal and hold alwayes a true limited, exact proportion one to another, but the arch and its chorde never can be equal, i. e. there never can be equality between a right-line and a crooked-line, both posited between the same two ter­minateing points, nor any analogy between the rectitude of the one and the curvature of the other. And the seeking to prove the equa­lity of angles contain'd under homologal lines, the one curve, the other a right-line from your usual fancy of a regular polygone of infinite angles in every circle, is too wild to be per­swasive: for though at any mean point in curve lines, the two parts of the curve-line may be conceived specially to meet as several parts and lines, and so to have inclination the one to the other, and so to constitute an angle; which we call the angle of curvature and co­incidence, not reasonably to be denyed by those, with whom it is so ordinary to make such suppositions, and especially such as can so usually against possibility imagine angles in a right-line remaining a right-line: yet that angles should be without sides, and a peri­meter of any figure conceived at once to be all angular points and no lineary sides, clearly [Page 214] [...]stes the perimeter of the nature of a line: [...] to me it seems far from the nature of a [...]ular figure, that hath nothing but points [...]stead of lines to bound it: but which is most material; that a number actually infinite should be so easily given, is hard to allow: and that indivisibles, as points, should be so adjacent [...] to another, one without another, with­ [...] coincidence, identity and unity, is new [...]osophy, and not easily capable of any in­ [...] defense. Therefore that argumentation [...], that such a regular polygone of in­ [...]whether sides or angles is either a circle, [...] inscribable in a circle, is too vain: for it can be neither, being nothing, because there neither is, nor can be any such thing: for if any such were allowed they must of necessity have equal and infinite perimeters; which is too gross to be admitted in it self, and besides renders the whole matter unapplyable to Cir­cles, which are acknowledged, to be some less then others. So as all discourses of a re­gular polygone of infinite angles, are discour­ses not only of a non-entity, but an absolute impossibility, which renders all suppositions thereof unjustifiable. And of the same fineness are those sayings, that the magnitude of an angle is not to be judged of from the divarica­tion which the sides have without the angular point, or point of concurrence, but from the diva­rication [Page 215] which they have in the point of con­currence; as if in an indivisible point they could have any divarication at all. But as if it were resolved that even this should be transcen­ded in monstrosity, for the justifying of the equality of mixed lined angles contain'd by homologal sides in unequal Circles, by an in­stance from the coapting of unequal hexagones to the same line, as a common side in them all, divided equally by a perpendicular passing thorow the centers of all, a right-lined angle is strangely constituted either of three right-lines concurring, but not in the same point, or of two lines without any concurrence, or else the instance must be void of all pertinency to the question. So to all those objections seem­ingly founded upon that proposition, or po­stulate, that what is less then any positive quantity whatsoever is not any quantity at all, is justly answered; that the proposition or postulate is most true and reasonable, and can­not by any of sound mind be denyed, or doub­ted: but no force of objection could be made out of that, if other things of a less veritable nature had not been taken in; as in most of them the fancyed possibility of a regular poly­gone of infinite angles; and frequently that a Circle is that regular polygone.

But besides, though what is less then any positive quantity whatsoever be not any quan­tity [Page 216] at all; yet this hinders not, but quanti­tyes may be mathematically heterogeneal and improportionable one to another: so every surface is less then any solid: and angles of Contact are not less then any quantity what­soever, for there is in the least of them an endles, unexhausted divisibility; which how it can consist with a nonquantitativenes, let those that have a mind to be serious solemnely consider. To the objection that would prove, neither semi-circumference to contain an angle with the right-lined tangent of it in its extreem point, because the two semi circumferences contain no angle at that point, but are one re­gularly continued line, and the circumference and right-lined tangent are lines coincident, at least as to the point of Contact; manifest and reasonable answers cannot be to seek out of what hath already been said. For first what hinders the reasonable conceiving of angularity at any point of a curve-line, where is both con­currence, inclination and divisibility, more then the notion of divisibility at any mean point of a right-line? And not to doubt but a curve line may be conceived reasonably as one continued line, as well as two, or more, in­clined and concurring right-lines; yet that the right-line tangent and curve-line which it toucheth, should be said to be coincident lines, in such sense as to exclude angularity; [Page 217] or that any two lines can be so coincident in one only point, as to exclude angularity, and the inflexion of one to, or from the other, except both lye in one and the same right-line; hath, as elsewhere, been plainly and abun­dantly answered to. To the objection that the Area of a Circle is equal to a rect-angle under the semidiameter and semi circumference; and that therefore the semidiameter in a Circle is perpendicular to the circumference in a Circle and makes at the circumference four equal right-angles; is answered, that the whole ob­jection is a manifest paralogisme. For it is not denyed but in the right-lined rect-angle under the semidiameter, and a right-line equal to the semi circumference, is presumed, and by the definition of a rect-angle inferred, that the an­gle under those two right-lines is a right right-lined angle: besides it is not denyed, but the diameter falls perpendicularly in the circle upon the circumference: and that the four an­gles made by the falling of the semidiameter up­on the circumference differ from one another less than the least right-lined angle: however that cannot force the falling of the semidiameter perpendicularly upon the circumference into the properties of perpendicularness between right lines, which still divides the space at the angular point into four angles always, every way alike, and equal; which in right lines per­pendiculars [Page 218] upon curve lines in the same plane, is impossible to be, and therefore impossible e­ver to be demonstrated.

It will not be unuseful here to enquire where­in the likeness and unlikeness of angles doth consist, and whether there be any such thing as likeness and unlikeness in angles, or whether the likeness, or unlikeness of figures be only in the similitude, or dissimilitude of the sides. And that by a circumspect consideration of the nature of mathematical similitude in other cases, we may be the better guided into the true and most rational notion of similitude in angles, let us remember what hath already been judged in this point, and what is herein confessed on all hands. First, in right lined figures, those fi­gures are judged like, whose answering angles are equal, and the answering sides and other lines proportionable; and if they be equal, they may be coapted, homologal side to homologal side, and answering angle to answering angle; or whether they be equal, or unequal, all the sides and other answering lines of the one may be set, as from the same center, each at paral­lelisme, or coincidency with the answering sides, or lines of the other, so as in like right lined figures is proportionablenes▪ in the answe­ring sides, equality of the answering angles, coaptability of all the answering sides into ei­ther coincidence, or equidistance, and a pro­portionate [Page 219] distance of the answering angles, each from the other. But now the similitude of figuration which is in circles founded upon the like genesis of all circles is in the equidi­stance, or coincidence of their circumferences, when the center of the one is coapted to the cen­ter of the other; and that equal angles from the center intercept proportionable parts of the cir­cumferences, and that proportionable parts of the circumferences are connected by proportio­nable chords, and contain and sustain equal right lined angles. And the like speculations might be pursued in other figures both plane and solid. In a rational application of which to the disquisition of angles, it may be first en­quired, whether there be any such thing as si­militude & dissimilitude to be own'd or observ'd among angles; and if so, how that similitude is to be understood; and whether it be inconsi­stent with inequality in the answering angles. To clear all which we must know, that in une­qual but like right lined figures, the homolo­gal angles are always equal, being contained in both figures under right lines: but in une­qual, and like mixed lined and crooked lined fi­gures, the homologal mixed lined, or crooked lined angles, neither are, nor can be equal; only their difference is ever less than the least right lined angle; and their similitude hath ne­ver rationally yet by any been questioned but [Page 220] with good reason according to the following gloss is to be justified. The more clearly to demonstrate all which in Fig. 21. upon the common center A. draw two unequal Circles, Viz. HEG. the lesser, and BCD. the greater. Then from any point B. in the greater circle BCD. draw the right line BEAFK. thorough the common center A. cutting the circumfe­rence of the lesser circle HEG. in the point E. then take AF. equal to BE. and upon the cen­ter F. and semidiameter FE. draw another cir­cle ECKD. equal to the greater circle CBD. Here on all hands is agreed, that the lesser cir­cle HEG. and the greater circle CBD. are like figures, and that therefore the two mixed lined recto-concave angles ABD. and AEG. are like angles. And by the construction it is apparent that the two recto-concave angles ABD. and AED. are equal: and that the two recto-con­cave angles AEG. and AED. are unequal: and that the recto-concave angle AED. is greater than the recto-concave angle AEG. and in all like cases it is always so; however the difference of the two angles must necessarily be less than any right lined angle, because all such citradiametral concavo-convexe angles of contact, as GED. are always less than any right lined angle; as is consequent to what hath been demonstrated in Geometry, which was to be shewn. Whence we may clearly observe [Page 221] that similitude of Figures lies chiefly in the pro­portionality and like positure of homologal sides, in respect of parallelisme and coincidence, without imposing any other necessity for the e­quality of answering angles, then as it may consist with the proportionating and like posi­ting of the homologal sides and lines. And such inequality of the answering angles, as is requi­site to the proportionating and alike positing of the homologal sides and lines in like and une­qual mixed, or crooked lined figures, is so far from being inconsistent with their figurative si­militude, that they cannot without it, under in­equality keep similitude in their figuration. And though the inequality, which is between angle and angle be less then that which is, or may be, between the Homologal sides and lines; yet the inequality of the angles is more different, being an inequality without proportion, where­as the inequality of the homologal sides and lines is ever according to proportion. Upon the whole, it is not equality that generally makes angles to be like, for a right lined, and an Iso­clitical concavo Convexe may be equal angles, but never can be like, nor were ever suspected to be so: but that which makes angles to be like is rather their being contained under homologal sides, posited so as to construct a like & homolo­gal figuration. And this whole matter depends upon what I before hinted, Viz. the figuration of [Page 222] lines and angles; Sc. rectitude, being one single, simple, figuration of lines incapable of any variety, like angles under right lines are always equal, and never can be unequal: but [...], being infinitely variable, those [...] are said to be like, i. e. homolo­gal, [...]hose construction is like, so as in like figure [...] upon a common center, to set homolo­gal sides and lines proportionably equidistant, or coincident; as circumferences of like though unequal circles, ellipses, &c. And so under a thousand inequalities such mixed and crooked lined angles may be like: as in Fig. 21. the recto-concave angles AEG. and AED. being unequal, are both like to the angle ABD. and so is every angle how different soever, if con­tained under a diameter and a circumference. And indeed the figuration of angles, being in­compleat, and the length of their sides unde­termined, neither parallelisme, nor coinciden­cy, nor proportionality, nor homologal posi­ture, can, when they are unequal, be conceived in their sides, without special relation to some compleat Figure and its Center: so the recto-concave angles AEG. and ABD in Fig. 21. are like, as conceived to be each contained re­spectively under a diameter and a circumfe­rence, and so upon a common center positable into parallelisme, coincidence and proportio­nableness, and all possible likeness and homo­logalness [Page 223] of figuration. In right-lined angles, where homologal and like angles are always e­qual▪ for the same reason every angle equal to a right-lined angle, is not presently a like an­gle: [...] a thousand equal angles, are all, ever, and to all purposes, unlike; as two equal iso­clitical right lined and crooked lined angles, be­cause they can never be coapted to be answering angles, in like Figures, or to set their contain­ing sides homologally, and in parallelisme, or coincidence. That equality of answering angles is not so of the essence of like Fi­gures, as proportionality of sides, and answer­ing lines, with their parallelisme, or coinci­dency: only from the propriety of like plane Figures, follows an equality in all like right-lined angles; and in like curve lined Fi­gures, that their inequality is ever less than the least right-lined angle. Hence therefore ap­pears that from the similitude which is in une­qual circles, the equality between angles of semi-circles & right right-lined angles is not effectu­ally proved. And notwithstanding any thing in those arguments tendred and proved, every recto-concave angle contained under a concave arch of a circle and a right-line, which is per­pendicular to the right line tangent of the arch at the angular point, is greater than any right-lined acute angle, and less than a right right-lined angle: and the recto-convexe angle con­tained [Page 224] under the convexe arch and the right lined tangent is less than any right lined angle whatsoever: and the other recto-convexe angle contained under the convexe arch and the right line, which is perpendicular to the right line tangent at the angular point, is greater than a right right-lined angle, and less than any ob­tuse right-lined angle whatsoever. And where­as you object that if, as in Fig. 12. the right line KGA. be the diameter of the circle KAD. and AB the right line tangent, then KAB is a right right lined angle: and the recto-convexe angle of contact DAB is no part of the right right lined angle KAB. that therefore the an­gle of the semicitcle KAD is still equal to a right right lined angle, because what is taken out of it was no part of it. I answer, the recto-convexe angle of contact DAB is indeed no proportionable part of the right right-lined an­gle KAB, but yet it is truly a part, though improportionable, and so mathematically he­terogeneal: for if it had been no part at all, and nothing, then the angle of the semicircle KAD. (nothing being taken out of the right right-li­ned angle KAB, but the recto-convexe angle of contact DAB, which is said by you to be nothing, and no angle) it should [...]ill remain a right right-lined angle; which is not by any asserted, the contrary being so manifest: be­sides that the separability of the recto-convexe [Page 225] angle DAB from the recto-concave angle KAD makes clear and certain, the truth of its being a part of the right-lined angle KAB.

And likewise from what hath been before declared in our opening the nature of a plane angle may clearly appear, that we are not to understand that a plane angle is meerely the an­gular point, or meerely in the angular point, as contradistinguished from the containing sides, though it there terminate, or thence have its rise; but angles are in the habitude of the concurring, containing and inclined sides: Viz. the habitude which they hold each to other all along their tendency unto the an­gular point, or their rise from thence, if we would have the full notion, inclination and figuration of an angle. For there is often a great inequality and vast imparity between the in­clination, sometimes of one part of the con­taining side to the other containing side, and the inclination thereunto, of other parts of the same first containing side; as may appear in all mixed lined, mixed crooked-lined and all other anisoclitical angles. And the nature of an angle consisting in inclination as well as in concur­rence, though concurrence may be and is in a point, and inclination at a point, yet incli­nation must be in the lines and of the lines, and cannot be in a point separately. And methinks the nature of an angle, and its inclination, is [Page 226] scarcely so fully held forth, when the inclination of the two lineary sides containing it, as if the sides were not therein concerned, is ordered to be observed only in the angular point, and not out of it: because, as you say, though you urge it to the contrary, many times out of the angular point in the containing sides, no two points can be shewen in the one side, where it hath the same inclination unto the other. Certainly in an indi­visible, such as is the angular point, if abstractly considered, it were vain to expect and impossi­ble to observe any inclination: and no doubt as the magnitudes inclined are without the angular point, so is also the inclination: though as they terminate in the angular point, so doth the incli­nation. So the angle of a semi circle is not the common terme of the diameter and the semi cir­cumference, excluding the diameter and the semi­cir-cumference for then in an abstracted point it should be possible to observe an inclination; and a point being indivisible should be inclined unto its self; which is not convenient to assert: but rather the angle of a semi circle is the in­clination of the semi circumference to the dia­meter, terminated in the angular point, which is common to both. And whereas it is said that out of the angular point, no two points can be shewen in the diameter, at which the diame­ter is equally inclined to the semi circumfe­rence, nor in the semi circumference where it [Page 227] is equally inclined to the diameter; all is allow­ed and averred as glosseably true, and this is that which makes the great difference be­tween anisoclitical and isoclitical angles, and renders it so impossible to give an anisoclitical angle equal to an isoclitical angle. For in iso­clitical angles the inclination of the sides, the one unto the other, is at all points the same without any variation; as every where appears by the interjected arches of circles drawn upon the angular point as Center: but in anisocli­tical angles, at every several point, the one containing side hath several and different incli­nations to the other containing side: which is the cause that isoclitical angles may possibly and easily be given, sometimes greater, and sometimes lesser then anisoclitical angles, but never equal: because in the one the inclina­tion of the containing sides still varieth, in the other not at all. And if the whole nature of an angle lye in the angular point, without exten­ding the habitude of it farther into the produ­ction and figuration of the containing sides; it will not only be necessary for us to yield unto you that recto-convexe angles of contact are not quantitative; but besides both you and we, contrary to what we have alwayes hitherto judged, shall be constrained to acknowledge that there is no quantitativenes neither in crooked-lined, nor right-lined, nor any other [Page 228] angles whatsoewer, whether superficial, in one, or several plaines, or solid. And can any thing be more horrid then to say, the quantity of angles is not to be measured by the divarica­tion of the sides at the angular point, but by their divarication in the angular point, where they have none at all? But yet though it is thus evident, that the inclination of the sides at the angular point, may and frequently is much less or greater then the inclination of the same sides at other points; which, as is above hinted, is not to be left out in the full, genuine and clear consideration of the nature of angles, their kinds, figurations and quantities: however the inclination of the sides at the angular point, is that which is most usually enquired after, and most useful to be searched and observed in Geometry; for the discoveries which are from thence made of lines, how they fall coinci­dently, or within, or without others. To the objection that in fig. 12. the right-lined tangent AB and the arch AL make all one and the same equal inclination to the right-line secant AC in the common angular point A; and that therefore the right-lined angle BAC under the right-lined secant AC and the right-line tangent AB is equal to the mixed-lined recto-concave angle EAL under the right-line secant AC and the arch AL; I answer as before, in­clination is not in the angular point abstractly [Page 229] considered without regard to the sides passing out of it, but inclination is the relative situa­tion which the concurring sides have at the angular point: at least that is their inclination there: for a point to a line can have no incli­nation: it may have distance from the line, but cannot be inclined unto it, because of its indivisibility. And having already shewn the inclination of side to side to be of the essence, notion and nature of an angle; a little may be reply enough to all those hypersceptical ob­jections which are sounded upon the imagi­nation of an angle in a right-line, or any incli­nation, or angularity imagined between a right line and a point, especially the point being in the right-line. And equality, or inequality of angles is not, nor can be judged of by the so abstractly considered angular point; in which a thousand several sides of several and unequal angles may meet indifferently; but the judgment of the magnitude and equality and inequality of angles is from the sides, and the order of the divarications, in which they passe, especially first of all, from the angular point. Besides how strangely is it taken for granted, without pro­ving, th [...] the right-line tangent AB and the arch AL are equally in the angular point A in­clined unto the right-line secant AC? If that could be once proved, the concern of it would turn the scales of the controversy: but demon­stration [Page 230] is so clear to the contrary, that as without proof it is not fit to be admitted, so for the proof of it, I know nothing can be pro­duced besides an utter despair of ever making it out.

For if the congruency of the sides termina­tively in the angular point, were sufficient to constitute equality in angles, it appears not how any angles meeting in the same or different angular points could be unequal: every point by reason of its indivisibility being incapable of inequality, as well as inclination.

And if all such angles so constituted by the falling within, or without of the sides, shall be doubted and questioned whether they be true and quantitative angles, and whether the addition, or subduction of them be able to diversify other angles and their quantities; all the pains of the Geometricians to prove the intracadency and extracadency of the angular sides from the same angular point, were vain and to no purpose; the angles remaining alto­gether the same and equal, whether such an­gles of contact be added to them, or taken from them. But yet though the true, [...] and ge­nuine nature of an angle consist in the mutual habitude and inclination of the containing and concurring sides, it is not ever necessary to con­sider it with such a largenes in Geometry. The inclination which the sides bear mutually each [Page 231] to other at the angular point is out of doubt that which is of most constant necessity, highest concern and usefulnes in all angles to be obser­ved. It is true in isoclitical angles, the incli­nation of the two containing sides being every where the same and equal, it is indifferently by Geometricians taken by a circle whose center is in the angular point, of what diameter soever thereunto applyable, and at what same distance soever from the angle, or at what same longitude soever from thence in the sides the points be, at which their inclination is ob­served by intercepted arches: and the same two points, terminating the intercepted arches at which their mutual inclination is observed, constantly offer themselves together, whether you take points at equal distance from the an­gular point, or intercepting in the sides equal longitudes between them and the same angular point But in anisoclitical angles; the inclina­tion of the anisoclitical sides varying still in the continuity of their production, if we use the former method of measuring the inclination of the sides by intercepted arches of a circle drawn upon the angular point as center; in them therefore Geometricians concern themselves little further then to observe the mutual incli­nation of the sides at the very point of their angle: and not at any other points in the ani­soclitical sides, save only the point of their [Page 232] concurrence; because of the constant varia­tion of their inclination, both in respect of such, and every other method and way of measuring, according to the continuity of their production. And by arches of circle drawn upon the angular point as center, it is impos­sible to measure the inclination of the anisocli­tical lines at the point of their concurrence; the only way therefore which remains unto Geome­tricians to measure such anisoclitical angles, i. e, the inclination of their sides at the point of their concurrence, is by observing the lines in what order they depart from the angular point: Viz. which line falls within, which without, and which is coincident with that unto which it is compared. So in fig. 12. if AHF and ADK be equal circles touching in the point A: and G the center, and AGK the diameter of the circle ADK: and AEL the arch of a greater circle touching both the former circles in the same point A and with its concave side at A respecting the center G, and with its convexe side at A re­specting the circle HAF: also if AB be a right-line tangent touching all the three former cir­cles in the point A, and the right-line ADEC cutt the circle ADK in the point [...], and the circle AEL in the point E: here the Geome­trician demonstrates the angle KAB to be a right right lined angle, and that the arch AFH passeth out of the angular point A without and [Page 233] beyond the right-line tangent AB, and with­out the right right-lined angle KAB: and that the arch AEL passeth from the angular point A within, or on this side of the right-line tan­gent AB, and within the right right-lined angle KAB, but without the arch ADK, and with­out the angle of the semi circle KAD: and that the right-line ADEC passeth from the an­gular point A within the arch ADK, and within the angle of the semi circle KAD. And this is all which is intended in Geometry when recto-convexe and such like angles of contact are said to be less then the least right-lined angle: not that there is any proportion between any angles of contact and right-lined angles; for there is none: but that the sides of any such angle of contact are coaptable, both of them within any the least right-lined angle. In like manner when one angle of contact is said to be greater or lesser then another; it is not the in­tendment of the Geometrician to assert any pro­portionablenes between them, but only to set down in what order the inclined lines pass, each in respect of the rest, from the angular point, or point of concurrence: and how each is conseque [...]ly more, or less inclined in that respect, to any one of the rest, though by the whole kind and without proportion, and with­out a common way of measuring their quanti­ties in some indefinite quantity, according to [Page 234] the kind of the measuring, indefinite quantity▪ the semblablenes of saying, one angle of con­tact is greater, or less then another being only in this, that when one side is coapted, the other falls within, or without; or when neither sides can possibly be coapted, both fall within, or both fall without, so as the one is really greater, or less then the other, not in propor­tion, but by the whole kind and heteroge­neally, as a two foot right-line is lesser then a four foot square, and a two foot right-line cannot be contained in an inch square, though in an inch square may be drawn a line ten thou­sand times longer, only out of rectitude. And to prove such an heterogeneal inequality, whether of the greater, or of the less between angles, is that which is frequently sufficient for many Geometrical purposes.

Only two things now remain of all your ob­jections and scrupulous quaeries. Viz.

First, why in some quantities, between the homogeneals, to a given quantity you cannot give another in what proportion you please greater or less, The answer is clear out of the foregoing discourse: because some quantities have but a limited extensibility: and other quantities have but a limited divisibility. All proportions at pleasure with respect to any given quantity, are only assigneable in those homo­geneals whose quantityes have both an unlimi­ted [Page 235] extensibility and an unlimited divisibility; either of which failing, though they cannot be, save in a limited proportion one to another, and to every given homogeneal, yet they can­not be in any proportion, at pleasure, to any given homogeneal.

And ^2ly. for your other Quaery; whether this controversy about the recto-convexe angle of contact concern abstract Mathematicks, whose glory used to be that it was devoid of Contro­versy, locking forth and keeping out all un­peaceful bickerings with the diamond-key of doubtless demonstration; or whether it concern concrete Mathematicks, which for its concre­tion unto matter cannot so well free its self from the intanglements of doubts and disputes? and if it concern the more abstract part of the Ma­thematicks; what was the first slip that drew so unworthy a disparagement upon that most noble piece of learning? In the preceding dis­course you may observe, I have been free and clear in my judgement concerning this; that it is a controversy of the purer and more noble part of the Mathematicks: and therefore of the higher concern, that it be brought to a fair decision and irrefragable diremption: and that the first spark from whence all this heat after­ward arose, was at first struck out of the du­biousnes and aequivocation of the word homo­geneity, mentioned in certain Mathematical [Page 236] definitions and Postulates; but of it self no where expressely defined in the Mathematicks; ex­cept we shall take one of the Pustulates, for its definition cast into the form of a Postulate; or as cryptically implying that definition and the postulate to be thereupon immediately formed: especially when they compared homogeneity in a mistaken sense, with some mathematical conclusions, which they had observed to be fairly demonstrated.

Pardon this my zeal on the behalf of the old learned Romane professour. And if these pa­pers seem long, remember yours were not short; and the question hath long troubled the world: and in matters that are new, and scarce yet well understood, to inculcate once and again the same things, is not only justify'd as allowable, but judged expedient and in such cases necessary. If you please, let the whole passe for a fuse Comment upon this; that in recto-convexe angles of contact is the inclina­tion required for an angle by the definition of an angle; and therefore they are angles and their sides not coincident: and they are angular parts of acknowledged angles separable from the remaining angular parts; and there is an in­finite divisibility in them, so as they must be quantitative: and yet it is demonstrated that they are improportionable to all right-lined angles; therefore they are heterogeneal: which [Page 237] inference is the more necessary, when upon examination we find nothing so convenably to set forth unto us the nature of mathematical ho­mogeneity, as the homometricalnes and the thence arising proportionablenes of the magni­tudes, and of mathematical heterogeneity as their heterometricalnes and improportionable­nes.