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. 4 th though AB and AC be supposed to be of uniform answering and equal curvatures and likewise in fig. 2 d. though AB and AC again supposed to be of uniform answering and equal curvatures and in fig. 3 d. fig 5 6 9 10 11. supposing AB and AC not to be of 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 7 th 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 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 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 B AF are not unequal and that neither of them is quantitative and that the crooked-lines E A 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 B AG is equal to the mixt-lined i. e. Recto-concave angles of the Semicircles E AG and D AG severally and that those angles of the Semicircles E AG and D AG are equal the one unto the other and that the mixed lined Recto-convexe angle F AG 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 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 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 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 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 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 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 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
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 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-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 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 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 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 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 S c. dec is equal to the right-lined angle hag S c. 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
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 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 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 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 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 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 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 C AH 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 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-crooked-lines ABC and AFH is equall to the respective arch EI intercepted between the two isoclitical right-right-lines and this wheresoever the point I be taken in the right-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 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 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 Dodeâ—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 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
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 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 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 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 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-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 S c. 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 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 â—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
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 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 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 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 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 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 â—hose construction is like so as in like figureâ— 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 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 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 KG A. be the diameter of the circle K AD. 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 D AB which is said by you to be nothing and no angle it should â—â—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 angle D AB 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 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 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
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 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 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 D AF and the arch BAC but only that GH being a right line and touching the arch B AC in the point A that I say the right line 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 B AC the contrary of which is above-shewn and confessed and besides hereupon should two arches B AC 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 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 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 t is 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 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-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 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 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 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 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 H AS therefore the ultradiametral concavo-convexe angle of Contact D AH is equal to two right right-lined angles and therefore the ultra-diametral concavo-convexe angle of Contact D AR 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 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
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 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 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 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 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-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 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
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 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 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 proâ—e Mathematical homogeneity the mensurablenes of the quantity of the compared magnitudes in the same indefinite quantity or measure for 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 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 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 speeâ—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
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 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 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 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 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 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
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-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 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 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 F AG 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 F AB and B AD 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 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 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 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
seldome are we are not to maintain Controversy as if they were the way of nature This Sir I have written not to Inform but confirm your judgement which I know so well versed in the Syntaxe of this our humane body that it cannot dissent from what is The Fifth Answer That immense volatility may consist with immense ponderousness That tincturs may be altered by maturation without any addition Whether the appearing and motion of comets may be before their appearance predicted That no such particular predictions can be made concerning the meteors which are in the Atmosphear of the earth nor of the first appearances of such Comets as are supposed to have their Original from new amassements of Cometical matter in other Atmospheares though after their first appearance upon some observations accurately made somewhat though nothing so peremptorily as in other cases may be predicted relating to their future motion That it is not impossible but concerning Comets which are permanent bodies and not new amassements predictions may be made long before of their future appearances and motion SIR I Take your satisfaction upon my last proposed clear and doubtless Experiments now since by you proved and approved as a fair acknowledgment of that Truth which however to me upon its former evidences needed no farther confirmation Viz. that so different are the genuine notions and qualities of fixednes and gravity that immense volatility may and is ordinarily consistent with an immense specifick ponderousness arising not from the moles and quantity of the bodies under consideration but rather from their natures and kinds That what in your first velitation you assumed as absurdly grosse inconsistent and impossible is now upon your own acknowledgments most easily and obviously demonstrable by experience No less certainly is to be acknowledged in what you propose for the tingeing of metalline bodies only by ordering and attending them in the management of their maturations without the addition either of body or spirit as being all the time of this operation under the sure seal Could I perswade my self it should not be resented as a grand unkindnes to be silent in what you call for my thoughts in in the close of yours Viz. Whether prognosticks at certainty such as are of Eclipses of Coitions oppositions motions of other Starrs whether I say such prognosticks and of like certainty may not also be made of Comets of their Appearances common impediments removed and of their motions I should if it might be herein lay Harpocrates finger upon my lip and seal them up into a pertinacions silence Not but I am desirous to know and willing to search after truth only I fear me these are secrets of nature by their peculiar mysteriousnes sinking themselves so low into the pit of obscurity that the stock of observations and disquisitions about them which the world as yet hath is not able to raise them so high and place them so near day as to be within humane reach and discovery What I now offer is a Caesarean birth of the mind not brought forth by me but cut out of the womb by your importunity if it want shape licking and lineaments accept it as an unripe Abort and either hatch it to perfection in your Thigh or give it a little dust to cover it I take by way of praesumption that in this Quaery and Question you mean not by Comets any of those more usual less permanent is regularly moved Meteors bred gathered fired and burning in the Atmosphaere of our Earth after their Appearance there cannot be much certainty of their Motion though sometimes there may be conjectures probable enough and in the Event by observation justifyed when the fuel or fovent matter of such Meteors is manifestly upon what account soever known to be only or most copiously situate and disposed some one particular way but of the generation and first appearance of such Meteors particularly the moment when and the point or exact place where they shall appear much less of certainty in such pronosticks is to be expected there being so much variety contingency and uncertainty in the causes and meeting together of those causes which contribute to their production And though there are and have been many praedictions in general and Rules of praedicting Astrologically or Physiologically from the seasons of the year the temperature and distemper of seasons c. Concerning such meteors in general yet in a matter so unallyed unto certainty replenished with all manner of casualties to promote or retard such productions I have not known any offer peremptory Pronosticks of the kind of the meteor its shape magnitude duration motion with absolute determinations of its time and places at any time beforehand The quality and condition of the subject matter making it as impossible to bolt out scientifical and oracularly certain predictions of such meteors as it is a year before to prove or shew that in such an hour and in such a quarter shall be a Rainbow so colour'd so continued or discontinued and of such limited dimensions or that such a moment from such a point of such an Azymuth shall a devolant Star spring forth which in its fall shall run obliquely thorow such and such Azymuths and expire at such an height above the Horison or that such an hour in such a Longitude and Latitude shall in such Altitudet and positions and of such dimensions be seen by day four Suns or by night three Moons To lay aside therefore the consideration of these as supposed impertinent to our present purpose what may judiciously be concluded upon your Questions concerning those other Comets which lye without the compasse of this Earths Atmospaere If there be not two sorts of such celestial Comets there are at least two several and very different Hypotheses and notions under which they are considered by Artists and Artists of equal worth and fame order their reasonings some upon the one supposition some upon the other Viz. Some as if Comets were new made bodies amassed and gathered in some of the superior Atmosphaeres many of which are not without fair reason supposed to be in those vastly remote aethereal regions others as if Comets were coaeval to and neither less permanent nor more new then the rest of the Stars only seldome seen and when seen soon passing again out of sight by reason of the Line upon which their Center is moved and nothing as yet appears hindering the truth of the possibility and consistency of both these Opinions leaving it especially indifferent in the later Hypothesis to call such stars at pleasure by the name either of Comets or New-stars or rather seldome appearing Stars Comets upon the first supposition seem not to want some affinity with several especially of the more eminent meteors of our Atmosphaere yet allowing a vast difference between them in place proportion duration motion and the like circumstantials And as our Globe though in its self large is but a minute thing compared with many
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 eveâ— 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 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 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 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 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 D AF 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 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
other for neither can the whole right right-lined angle nor the recto-concave angle 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 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 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 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 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 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
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-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 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 it s 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 â—â—â—â—stes the perimeter of the nature of a line 〈◊〉 to me it seems far from the nature of a â—â—â—ular figure that hath nothing but points â—â—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 withâ—â—â— coincidence identity and unity is new â—â—â—â—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 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 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 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
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 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 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 thâ—â— 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 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 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 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