VIA REGIA AD GEOMETRIAM. THE WAY TO GEOMETRY. Being necessary and useful, FOR Astronomers. Geographers. Land-meaters. Seamen. Engineres. Architecks. Carpenters. Painters. Carvers, etc. Written in Latin by PETER RAMUS, and now Translated and much enlarged by the Learned Mr. WILLIAM BEDWELL. LONDON, Printed by Thomas Cotes, And are to be sold by Michael Spark; at the blue Bible in Green Arbour, 1636. TO THE WORSHIPFUL M. JOHN GREAVES, Professor of Geometry in Gresham College London; All happiness. SIR, YOur acquaintance with the Author before his death was not long, which I have oft heard you say, you counted your great unhappiness, but within a short time after, you knew not well whether to count yourself more happy in that you once knew him, or unhappy in that upon your acquaintance you so suddenly lost him. This his work then being to come forth to the censorious eye of the world, and as the manner usually is to have some Patronage. I have thought good to dedicate it to yourself; and that for these two reasons especially. First, in respect of the sympathy betwixt it, and your studies; Labours of this nature being usually offered to such persons whose profession is that way settled Secondly, for the great love and respect you always showed to the Author, being indeed a man that would deserve no less, humble, void of pride, ever ready to impart his knowledge to others in what kind soever, loving and affecting those that affected learning. For these respects then, I offer to you this Work of your so much honoured friend. I myself also (as it is no less my duty) for his sake striving to make you hereby some part of a requital, lest I should be found guilty of ingratitude, which is a solecism in manners, if having so fit an opportunity, I should not express to the world some Testimony of love to you, who so much loved him. I desire then (good Sir) your kind acceptance of it, you knowing so well the ability of the Author, and being also able to judge of a Work of this nature, and in that respect the better able to defend it from the fury of envious Detractours, of which there are not few. Thus with my best wishes to you, as to my much respected friend, I rest. Yours to be commanded in any thing that he is able. JOHN CLERKE. To the Reader. FRiendly Reader, that which is here set forth to thy view, is a Translation out of Ramus. Formerly indeed Translated by one Mr. Thomas Hood, but never before set forth with the Demonstrations and Diagrams, which being cut before the Author's death, and the Work itself finished, the Copy I having in mine hands, never had thought for the promulgation of it, but that it should have died with its Author, considering no small prejudice usually attends the printing of dead men's Works, and we see the times, the world is now all ear and tongue, the most given with the Athenians, to little else than to hear and tell news: And if Apelles that skilful Artist always found somewhat to be amended in those Pictures which he had most curiously drawn; surely much in this Work might have been amended if the Author had lived to refine it, but in that it was only the first draught, and that he was prevented by death of a second view, though perused by others before the Press; I was ever unwilling to the publication, but that I was often and much solicited with iteration of strong importunity, and so in the end overruled: persuading me from time to time unto it, and that it being finished by the Author, it was far better to be published, though with some errors and escapes, than to be only moths-meat, and so utterly lost. I would have thee, Courteous Reader know, that it is no conceit of the worth of the thing that I should expose the name and credit of the Author to a public censure; yet I durst be bold to say, had he lived to have fitted it, and corrected the Press, the work would have pointed out the workman. For I may say, without vain ostentation, he was a man of worth and note, and there was not that kind of learning in which he had not some knowledge, but especially for the Eastern tongues, those deep and profound Studies, in the judgement of the learned, which knew him well, he hath not left his fellow behind him; as his Works also in Manuscript now extant in the public Library of the famous University of Cambridge; do testify no less; for him then being so grave and learned a Divine to meddle with a work of this nature, he gives thee a reason in his own following Preface for his principal end and intent of taking this Work in hand, was not for the deep and judicial, but for the shallowest skull, the good and profit of the simpler sort, who as it was in the Latin, were able to get little or no benefit from it. Therefore considering the worth of the Author, and his intent in the Worke. Read it favourably, and if the faults be not too great, cover them with the mantle of love, and judge charitably offences unwillingly committed, and do according to the terms of equity, as thou wouldst be done unto, but it is a common saying, as Printers get Copies for their profit, so Readers often buy and read for their pleasure; and there is no work so exactly done that can escape the malevolous disposition of some detracting spirits, to whom I say, as one well, Facilius e●t unicuivis nostrum aliena curiosè observare: quam proproia negotia rectè agere. It is a great deal more easy to carp at other men's doings, than to give better of his own. And as Arist. 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉; omnibus placere difficilimum est. But wherefore, Gentle Reader, should I make any doubt of thy courtesy, and favourable acceptance; for surely there can be nothing more contrary to equity, than to speak evil of those that have taken pains to do good, a Pagan would hardly do this, much less I hope any good Christian. Read then, and if by reading, thou reapest any profit, I have my desire, if not, the fault shall be thine own, reading haply more to judge and censure, than for any good and benefit which otherwise may be received from it; let but the same mind towards thine own good possess thee in reading it, as did the Author in writing it, and there shall be no need to doubt of thy profit by it. Thine in the common bond of love, JOHN CLERKE. The Author's Preface. TWo things, I fear me, will here be objected against me: The one concerneth myself, directly: The other mine Author, and the work I have taken in hand the translating of him. Concerning myself, I suppose, some will ask, Why I being a Divine; should meddle or busy myself with these profane studies? Geometry may no way further Divinity, and therefore is no fit study for a Divine? This objection seemeth to smell of Brownisme, that is, of a rank peevish humour overflowing the stomach of some, whereby they are caused to loath all manner of solid learning, yea of true Divinity itself, and therefore it doth not deserve an answer: And this we in our Title before signified. For we have not taken this pains for Turks and others, who by the laws of their profession are bound to abandon all manner of learning. But if any man shall propose it, as a question, with a desire of satisfaction, we are ready to answer him to the best of our ability. First, that Theologia vera e●t ●rs artium & scientia scientiarum, Divinity is the Art of Arts, and Science of Sciences; or Divinity is the Mistress upon which all Arts and Sciences are to attend as servants and handmaids. And why then not Geometry? But in what place she should follow her, I dare not say: For I am no herald, and therefore I meddle not with precedency: But if I were, she should be none of the hindermost of her train. The Orator saith, and very truly doubtless, That, Omnes arts, quae humanitatem pertinent, habent commune quoddam vinculum, & cognatione quadam inter se continentur. All Arts which pertain unto humanity, they have a certain common b●nd, and are knit together by a kind of affinity. If then any Arts and Sciences may be thought necessary attendants upon this great Lady: Then surely Geometry amongst the rest must needs be one: For otherwise her train will be but loose and shattered. Plato saith 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, That God doth always work by Geometry, that is, as the ●●iseman doth interpret it, SAP. XI. 21. Omnia in mensura & numero & pondere disponere. Dispose all things by measure, and number, and weight: Or, as the learned Pl●tarch speaketh, He adorneth and layeth out all the parts of the world according to rate proportion, and similitude. Now who, I pray you, understandeth what these terms mean, but he which hath some mean skill in Geometry? Therefore none but such an one, may be able to declare and teach these things unto others. How many things are there in holy Scripture which may not well be understood without some mean skill in Geometry? The Fabric and bigness of Noah's Ark: The Sciagraphy of the Temple set out by Ezechiel, Who may understand, but he that is skilful in these Arts? I speak not of many and sundry words both in the New and Old Testaments, whose genuine and proper signification is m●rely Geometrical: And cannot well be conceived but of a Geometer. And here, that I may speak it without offence, I would have it observed, how many men, much magnified for learning, not only in their speeches, which always are not premeditated, but even in their writings, exposed to the view and censure of all men, do often paralogizein, speak much, and little to the purpose. This they could not so easily and often do, if they had been but meanly practised in these kind of studies. Wherefore that Epigram which was used to be written over their Philosophy School doors, 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, No man ignorant of Geometry come within these doors: Now written over our Divinity Schools. And if any man shall think this an hard sentence, let him hear what Saint Augustine saith in the same case, Nemo ad divinarum humanarumque rerum cognitionēm accedar, nisi prius annumerandi artem addiscat: Let no man come neither within the Divinity nor Philosophy Schools, except he have first learned Arithmetic. Now that the o●e of these Arts cannot be learned without the other; Euclid our great Master, who made but one of both, hath sufficiently demonstrated. If I should allege the like practice of famous Divines, greatly admired for their great skill in this profession, as T. Peckham Archbishop of Canterbury, Maurolycus Bishop of Messana in Sicilia, Cusanus Cardinal of Rome, and many others● before indifferent judges, I am sure I should not be condemned. Who doth not greatly magnify the grave Seb. Munster, the nimble Ph. Melanchthon, and the noble Bernardino Baldo Abbot of Guastill, and the painful Barth. Pitiscus of Grunberg, for their knowledge and pains in these Arts and Sciences? And thus much shall at this time suffice, to have spoken unto the first Question: If any shall require further satisfaction, those I refer unto the forenamed Authors, whose authority peradventure may more prevail with them, than my reasons may. The next is concerning mine Author, and the work in hand Geometry, it must needs be confessed we are beholden to Euclides Elements for: And he that would be rich in that profession, may have, if he be not covetous, his fill there, if he will labour hard, and take pains for it, it is true. But in what time think yau, may a man learn all Euclid, and so by him be made skilful in this Art? By himself I know not whether ever or never: And with the help of another, although very expert, I will not promise him that he shall attain to perfection in many years. Hypocrates the Prince of Physicians hath, as they say, in his works laid out the whole Art of Physic; but I marvel how long a man should study him alone, and read him over and over, before he should be a good Physician? I fear me all the friends that he hath, and neighbours round about him, yea, and himself too, would all die before he should be able to help them, or peradventure ere he should be able to know what they ailed; and after 30, or 40. years of such his study, I would be very loath to commit myself unto him. How much therefore are the students of this noble Science beholding unto those men, who by their industry, practice, and painful travels, have showed them a ready and certain way through this wilderness? The Elements of Euclid they do contain generally the whole art of Geometry: But if you will offer to travel thorough them alone, you shall find them, I will warrant you, Elements indeed: for there you may walk through the spacious Air, and over the great and wide sea, and in and about the vast and arid wilderness many a day and night, before you shall know where you are. This Ramus, my Author in reading him found to be true; and confesseth himself often to have been at a stand: Often to have lost himself: Often to have hit upon a rock, when he had thought he had touched land. Lest therefore other men, in this journey do not likewise lose themselves, for the benefit and safety, I mean, of others he hath pricked them out a charred or chacked out a way, which if thou shalt please to follow, it shall lead thee to thy ways end, as directly, and in as s●ort time, as conveniently may be. Yet in what time I cannot warrant thee: For all men's capacity, especially in these Arts, is not alike: All are not a like painful, industrious, or diligent: All are not of the same ability of body, to be able to continue or ●it at it: Or all not so free from other employments or business calling them from their study, as some others are. For know this for certain, Thou shalt here make no great progress, except thou do make it as it were a continued labour, Here you must observe that rule of the great Painter, Nulla dies sine linea, Let no day pass over your head, in which you draw not some diagram or figure or other. One other thing let me also advise thee of, how capable soever thou art, refuse not, if thou mayst have it, the help of a teacher: For except thou be another Hypocrates or Forcatelus, whom our Author mentioneth, thou canst not in these Arts and Sciences attain unto any great 〈…〉 great loss of most 〈…〉, For they are therefore called 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 Ma●● 〈…〉 is, doctrina●or disciplinary Arts, because they are not to be 〈◊〉 unto by our own information and industry; but by the help and instruction of others. This Work gentle Reader, was in part above 30. years since published by M. Thomas Hood, a learned man, and loving friend of mine, who teaching these Arts, in the Staplers Chapel in. Leadenhall London, for the benefit of his Scholars and Auditory, did set out the Elements apart by themselves. The whole at large, with the Diagrams, and Demonstrations, he● promised, as appeareth in the Preface to that his Work, at his convenient leisure to send out shortly, after them. This for aught we know or can learn, is not by him or any other performed: And yet are those alone, without these of small use or none to a learner, where a teacher is not always at hand. Wherefore we are bold being (encouraged thereunto by some private friends, and especially by the learned M. H. Brigges, professor of Geometry in the famous University of Oxford) to publish this of ours long since finished and ended. The usual terms, whether Latin or Greek, commonly used by the Geometers, we have set down and expressed in English, as well as we could, as others, writing of this argument in our language, have done before us. These terms, I doubt not, may by some in English otherwise be expressed, but how harsh those terms, may unto Mathematical ears, at the first appear, I will not say; and use in short time will make these familiar, and as pleasing to the ear as those possibly may be. Our Author, in the declaration of the Elements hath many passages, which in our judgement do not make so much for the understanding of the matter in hand, as for the defence of the method here used, against Aristotle, Euclid, Proclus, and others, which we have therefore wholly ●●red. Some other things, which in our opinion, might in some respect illustrate any particular in this business, we have here and there inserted. Out of the learned Finkins' Geometria Rotundi, We have added to the fifth Book certain Propositions with their Consectaries out of Ptolomi's Almagest. The painful and diligent Rod● Snellius out of the Lectures and Annotations of B. Salignacus, I. Tho. Freigius, and others, hath illustrated and altered here and there some few things. The Contents. Book I. Of a Magnitude. Page 1 Book II. Of a Line. p. 13 Book III. Of an Angle. p. 21 Book IV. Of a Figure. p. 32 Book V. Of Lines and Angles in a plain Surface. p. 51 Book VI Of a Triangle. p. 83 Book VII. The comparison of Triangles. p. 94 Book VIII. Of the divers kinds of Triangles. p. 106 Book IX. Of the measuring of right lines by like rightangled Triangles. p. 113 Book X. Of a Triangulate and Parallelogramme. p. 136 Book XI. Of a Rightangle. p. 148 Book XII. Of a Quadrate. p. 152 Book XIII. Of a Oblong. p. 167 Book XIV. Of a right line proportionally cut: And of other Quadrangles, and Multangles. p. 174 Book XV. Of the Lines in a Circle. p. 201 Book XVI. Of the Segments of a Circle. p. 201 Book XVII. Of the Adscription of a Circle and Triangle. p. 215 Book XVIII. Of the adscription of a Triangulate. p. 221 Book XIX. Of the measuring of ordinate Multangle, and of a Circle. p. 252 Book XX. Of a Bossed surface. p. 257 Book XXI. Of Lines and Surfaces in solids. p. 242 Book XXII. Of a Pyramid. p. 249 Book XXIII. Of a Prisma. p. 256 Book XXIV. Of a Cube. p. 264 Book XXV. Of mingled ordinate P●lyedra's. p. 171 Book XXVI. Of a Sphere. p. 284 Book XXVII. Of the Cone and Cylinder. p. 290 VIA REGIA AD GEOMETRIAM. THE FIRST BOOK OF Peter Ramus' Geometry, Which is of a Magnitude. 1. Geometry is the Art of measuring well. THE end or scope of Geometry is to measure well: Therefore it is defined of the end, as generally all other Arts are. To measure well therefore is to consider the nature and affections of every thing that is to be measured: To compare such like things one with another: And to understand their reason and proportion and similitude. For all that is to measure well, whether it be that by Congruency and application of some assigned measure: Or by Multiplication of the terms or bounds: Or by Division of the product made by multiplication: Or by any other way whatsoever the affection of the thing to be measured be considered. But this end of Geometry will appear much more beautiful and glorious in the use and geometrical works and practise then by precepts, when thou shalt observe Astronomers, Geographers, Land-meaters, Seamen, Engineers, Architects, Carpenters, Painters, and Carvers, in the description and measuring of the Stars, Countries, Lands, Engines, Seas, Buildings, Pictures, and Statues or Images to use the help of no other art but of Geometry. Wherefore here the name of this art cometh far short of the thing meant by it. (For Geometria, made of Gè, which in the Greek language signifieth the Earth; and Métron, a measure, importeth no more, but as one would say Land-measuring. And Geometra, is but Agrimensor, A land-meter: or as Tully calleth him Dec●mpedator, a Pole-man: or as Plautus, Finitor, a Marksman.) when as this Art teacheth not only how to measure the Land or the Earth, but the Water, and the Air, yea and the whole World too, and in it all Bodies, Surfaces, Lines, and whatsoever else is to be measured. Now a Measure, as Aristotle doth determine it, in every thing to be measured, is some small thing conceived and set out by the measurer; and of the Geometers it is called Mensura flamosa, a known measure. Which kind of measures, were at first, as Vitruvius and Herodo teach us, taken from man's body: whereupon Protagoras said, That man was the measure of all things, which speech of his, Saint john, Apoc. 21. 17. doth seem to approve. True it is, that beside those, there are some other sorts of measures, especially greater ones, taken from other things, yet all of them generally made and defined by those. And because the stature and bigness of men is greater in some places, than it is ordinarily in others, therefore the measures taken from them are greater in some countries, than they are in others. Behold here a catalogue, and description of such as are commonly either used amongst us, or some times mentioned in our stories and other books translated into our English tongue. Granum hordei, a Barley corn, like as a wheat corn in weights, is no kind of measure, but is quiddam minimum in mensura, some least thing in a measure, whereof it is, as it were, made, and whereby it is rectified. Digitus, a Finger breadth, containeth 2. barley corns length, or four laid side to side: Pollex, a Thumb breadth; called otherwise Vucia, an inch, 3. barley corns in length: Palmus, or Palmus minor, an Handbreadth, 4. fingers, or 3. ynches. Spithama, or Palmus major, a Span, 3. hands breadth, or 9 ynches. Cubitus, a Cubit, half a yard, from the elbow to the top of the middle finger, 6. hands breadth, or two spans. Vlna, from the top of the shoulder or armhole, to the top of the middle finger. It is two fold; A yard and an Elne. A yard, containeth 2. cubits, or 3. foot: An Elne, one yard and a quarter, or 2. cubits and ½. Pes, a Foot, 4. hands breadth, or twelve ynches. Gradus, or Passus minor, a Step, two foot and an half. Passus, or Passus major, a Stride, two steps, or five foot. Pertica, a Pertch, Pole, Rod or Lugge, 5. yards and an half. Stadium, a Furlong; after the Romans, 125. pases: the English, 40. rod. Milliare, or Milliarium, that is mille passus, 1000 passes, or 8. furlongs. Leuca, a League, 2. miles: used by the French, spaniards, and seamen. Parasanga, about 4. miles: a Persian, & common Dutch mile; 30. furlongs. Schoenos, 40. furlongs: an Egyptian, or swedland mile. Now for a confirmation of that which hath been said, hear the words of the Statute. It is ordained, That 3. grains of Barley, dry and round, do make an Inch: 12. ynches do make a Foot: 3. foot do make a Yard: 5. yards and ½ do make a Perch: And 40. perches in length, and 4. in breadth, do make an Acre: 33. Edwar. 1. De terris mensurandis: & De compositione ulnarum & Perticarum. Item, Be it enacted by the authority aforesaid; That a Mile shall be taken and reckoned in this manner, and no otherwise; That is to say, a Mile to contain 8. furlongs: And every Furlong to contain 40. lugges or poles: And every Lugge or Pole to contain 16. and●½ ●½. 25. Eliza. An Act for restraint of new building, etc. These, as I said, are according to divers countries, where they are used, much different one from another: which difference, in my judgement; ariseth especially out of the difference of the Foot, by which generally they are all made, whether they be greater or lesser. For the Hand being as before hath been taught, the fourth part of the foot whether greater or lesser: And the Inch, the third part of the hand, whether greater or lesser. Item, the Yard, containing 3. foot, whether greater or lesser: And the Rod 5. yards and ½, whether greater or lesser, and so forth of the rest; It must needs follow, that the Foot being in some places greater than it is in other some, these measures, the Hand, I mean, the Inch, the Yard, the Rod, must needs be greater or lesser in some places than they are in other. Of this diversity therefore, and difference of the foot, in foreign countries, as far as mine intelligence will inform me, because the place doth invite me, I will here add these few lines following. For of the rest, because they are of more special use, I will God willing, as just occasion shall be administered, speak more plentifully hereafter. Of this argument divers men have written somewhat, more or less: But none to my knowledge, more copiously and curiously, than james Capell, a Frenchman, and the learned Willebrand, Snellius, of Leiden in Holland, for they have compared, and that very diligently, many and sundry kinds of these measures one with another. The first as you may see in his treatise De mensuris intervallorum describeth these eleven following: of which the greatest is Pes Babylonius, the Babylonian foot; the least, Pes Toletanus, the foot used about Teledo in Spain: And the mean between both, Pes Atticus, that used about Athens in Greece. For they are one unto another as 20. 15. and 12. are one unto another. Therefore if the Spanish foot, being the least, be divided into 12. ynches, and every inch again into 10. parts, and so the whole foot into 120. the Attic foot shall contain of those parts 150. and the Babylonian, 200. To this Attic foot, of all other, doth ours come the nearest: For our English foot comprehendeth almost 152. such parts. The other, to wit the learned Snellius, in his Eratosthenes Batavus, a book which he hath written of the true quantity of the compass of the Earth, describeth many more, and that after a far more exact and curious manner. Here observe, that besides those by us here set down, there are certain others by him mentioned, which as he writeth are found wholly to agree with some one or other of these. For Rheinlandicus, that of Rheinland or Leiden, which he maketh his base, is all one with Romanus, the Italian or Roman foot. Lovaniensis, that of Lovane, with that of Antwerp: Bremensis, that of Breme in Germany, with that of Hafnia, in Denmark. Only his Pes Arabicus, the Arabian foot, or that mentioned in Abulfada, and Nubiensis: the Geographers I have overpassed, because he dareth not, for certain, affirm what it was. Look of what parts Pes Tolitanus, the spanish foot, or that of Toledo in Spain, containeth 120. of such is the Pes. Heidelbergicus, that of Heidelberg, 137. Hetruscus, that of Tuscan, in Italy, 138. Sedanensis, of Sedan in France, 139. Romanus, that of Rome in Italy, 144. Atticus, of Athens in Greece, 150. Anglicus, of England, 152. Parisinus, of Paris in France, 160. Syriacus, of Syria, 166. AEgyptiacus, of Egypt, 171. Hebraicus, that of judaea, 180. Babylonius, that of Babylon, 200. Look of what parts Pes Romonus, the foot of Rome, (which is all one with the foot of Rheinland) is 1000 of such parts is the foot of Toledo, in Spain, 864. Mechlin, in Brabant, 890. Strausburgh, in Germany, 891. Amsterdam, in Holland, 904. Antwerp, in Brabant, 909. Bavaria, in Germany, 924. Coppen-haun, in Denmark, 934. Goes, in Zealand, 954. Middleburge, in Zealand, 960. London, in England, 968. Noremberge, in Germany, 974. Ziriczee, in Zealand, 980. The ancient Greek, 1042. Dort, in Holland, 1050. Paris, in France, 1055. Briel, in Holland, 1060. Venice, in Italy, 1101. Babylon, in Chaldaea, 1172. Alexandria, in Egypt, 1200. Antioch, in Syria, 1360. Of all other therefore our English foot cometh nearest unto that used by the greeks: And the learned Master Ro. Hues, was not much amiss, who in his book or Treatise De Globis, thus writeth of it Pedem nostrum Angli cum Graecorum pedi aequalem invenimus, comparatione facta cum Graecorum pede, quem Agricola & alijex antiquis monumentis tradi derunt. Now by any one of these known and compared with ours, to all English men well known the rest may easily be proportioned out. 2. The thing proposed to be measured is a Magnitude. Magnitudo, a Magnitude or Bigness is the subject about which Geometry is busied. For every Art hath a proper subject about which it doth employ all his rules and precepts: And by this especially they do differ one from another. So the subject of Grammar was speech; of Logic, reason; of Arithmetic, numbers; and so now of Geometry it is a magnitude, all whose kinds, differences and affections, are hereafter to be declared. 3. A Magnitude is a continual quantity. A Magnitude is quantitas continua, a continued, or continual quantity. A number is quantitas discreta, a disjoined quantity: As one, two, three, four; do consist of one, two, three, four unities, which are disjoined and severed parts: whereas the parts of a Line, Surface, and Body are contained and continued without any manner of disjunction, separation, or distinction at all, as by and by shall better and more plainly appear. Therefore a Magnitude is here understood to be that whereby every thing to be measured is said to be great: As a Line from hence is said to be long, a Surface broad, a Body solid: Wherefore Length, Breadth, and solidity are Magnitudes. 4. That is continuum, continual, whose parts are contained or held together by some common bound. This definition of itself is somewhat obscure, and to be understand only in a geometrical sense: And it dependeth especially of the common bound. For the parts (which here are so called) are nothing in the whole, but in a potentia or pour: Neither indeed may the whole magnitude be conceived, but as it is compact of his parts, which notwithstanding we may in all places assume or take as contained and continued with a common bound, which Aristotle nameth a Common limit; but Euclid a Common section, as in a line, is a Point, in a surface, a Line: in a body, a Surface. 5. A bound is the outmost of a Magnitude. Terminus, a Term, or Bound is here understood to be that which doth either bound, limit, or end actu, in deed; as in the beginning and end of a magnitude: Or potentia, in power or ability, as when it is the common bound of the continual magnitude. Neither is the Bound a part of the bounded magnitude: For the thing bounding is one thing, and the thing bounded is another: For the Bound is one distance, dimension, or degree, inferior to the thing bounded: A Point is the bound of a line, and it is less than a line by one degree, because it cannot be divided, which a line may. A Line is the bound of a surface, and it is also less than a surface by one distance or dimension, because it is only length, whereas a surface hath both length and breadth. A Surface is the bound of a body, and it is less likewise than it is by one dimension, because it is only length and breadth, whereas as a body hath both length, breadth, and thickness. Now every Magnitude actu, in deed, is terminate, bounded and finite, yet the geometer doth desire some time to have an infinite line granted him, but no otherwise infinite or farther to be drawane out then may serve his turn. 6. A Magnitude is both infinitely made, and continued, and cut or divided by those things wherewith it is bounded. A line, a surface, and a body are made gemetrically by the motion of a point, line, and surface: Item, they are contained, continued, and cut or divided by a point, line, and surface. But a Line is bounded by a point: a surface, by a line: And a Body by a surface, as afterward by their several kinds shall be understood. Now that all magnitudes are cut or divided by the same wherewith they are bounded, is conceived out of the definition of Continuum, e. 4. For if the common band to contain and couple together the parts of a Line, surface, & Body, be a Point, Line, and Surface, it must needs be that a section or division shall be made by those common bands: And that to be dissolved which they did contain and knit together. 7. A point is an undivisible sign in a magnitude. A Point, as here it is defined, is not natural and to be perceived by sense; Because sense only perceiveth that which is a body; And if there be any thing less than other to be perceived by sense, that is called a Point. Wherefore a Point is no Magnitude: But it is only that which in a Magnitude is conceived and imagined to be undivisible. And although it be void of all bigness or Magnitude, yet is it the beginning of all magnitudes, the beginning I mean potentiâ, in power. 8. Magnitudes commensurable, are those which one and the same measure doth measure: chose, Magnitudes incommensurable are those, which the same measure cannot measure. 1, 2. d. X. Magnitudes compared between themselves in respect of numbers have Symmetry or commensurability, and Reason or rationality: Of themselves, Congruity and Adscription. But the measure of a magnitude is only by supposition, and at the discretion of the Geometer, to take as pleaseth him, whether an inch, an hand breadth, foot, or any other thing whatsoever, for a measure. Therefore two magnitudes, the one a foot long, the other two foot long, are commensurable; because the magnitude of one foot doth measure them both, the first once, the second twice. But some magnitudes there are which have no common measure, as the Diagony of a quadrate and his side, 116. p. X. actu, in deed, are Asymmetra, incommensurable: And yet they are potentiâ, by power, symmetra, commensurable, to wit by their quadrates: For the quadrate of the diagony is double to the quadrate of the side. 9 Rational Magnitudes are those whose reason may be expressed by a number of the measure given. chose they are irrationals. 5. d. X. Ratio, Reason, Rate, or Rationality, what it is our Author (and likewise Salignacus have taught us in the first Chapter of the second book of their Arithmetickes: Thither therefore I refer thee. Data mensura, a Measure given or assigned, is of Euclid called Rhetè, that is spoken, (or which may be uttered) definite, certain, to wit which may be expressed by some number, which is no other than that, which as we said, was called mensura famosa, a known or famous measure. Therefore Irrational magnitudes, on the contrary, are understood to be such whose reason or rate may not be expressed by a number or a measure assigned: As the side of the side of a quadrate of 20. foot unto a magnitude of two foot; of which kind of magnitudes, thirteen sorts are mentioned in the tenth book of Euclides Elements: such are the segments of a right line proportionally cut, unto the whole line. The Diameter in a circle is rational: But it is irrational unto the side of an inscribed quinquangle: The Diagony of an icosahedron and Dodecahedron is irrational unto the side. 10. Congruall or agreeable magnitudes are those, whose parts being applied or laid one upon another do fill an equal place. Symmetria, Symmetry or Commensurability and Rate were from numbers: The next affections of Magnitudes are altogether geometrical. Congruentia, Congruency, agreeableness is of two magnitudes, when the first parts of the one do agree to the first parts of the other, the mean to the mean, the extremes or ends to the ends, and lastly the parts of the one, in all respects to the parts, of the other: so Lines are congruall or agreeable, when the bounding, points of the one, applied to the bounding points of the other, and the whole lengths to the whole lengthes, do occupy or fill the same place. So Surfaces do agree, when the bounding lines, with the bounding lines: And the plots bounded, with the plots bounded do occupy the same place. Now bodies if they do agree, they do seem only to agree by their surfaces. And by this kind of congruency do we measure the bodies of all both liquid and dry things, to wit, by filling an equal place. Thus also do the moniers judge the moneys and coins to be equal, by the equal weight of the plates in filling up of an equal place. But here note, that there is nothing that is only a line, or a surface only, that is natural and sensible to the touch, but whatsoever is natural, and thus to be discerned is corporeal. Therefore 11. Congruall or agreeable Magnitudes are equal. 8. ax.j. A lesser right line may agree to a part of a greater, but to so much of it, it is equal, with how much it doth agree: Neither is that axiom reciprocal or to be converted; For neither in deed are Congruity and Equality reciprocal or convertible. For a Triangle may be equal to a Parallelogramme, yet it cannot in all points agree to it: And so to a Circle there is sometimes sought an equal quadrate, although in congruall or not agreeing with it: Because those things which are of the like kind do only agree. 12. Magnitudes are described between themselves, one with another, when the bounds of the one are bounded within the bounds of the other: That which is within, is called the inscript: and that which is without, the Circumscript. Now followeth Adscription, whose kinds are Inscription and Circumscription; That is when one figure is written or made within another: This when it is written or made about another figure. Homogenea, Homogenealls or figures of the same kind only between themselves rectitermina, or right bounded, are properly ascribed between themselves, and with a round. Notwithstanding, at the 15. book of Euclides Elements Heterogenea, Heterogenealls or figures of divers kinds are also ascribed, to wit the five ordinate plain bodies between themselves: And a right line is inscribed within a periphery and a triangle. But the use of adscription of a rectilineall and circle, shall hereafter manifest singular and notable mysteries by the reason and means of adscripts; which adscription shall be the key whereby a way is opened unto that most excellent doctrine taught by the subtenses or inscripts of a circle as Ptolomey speaks, or Sines, as the latter writers call them. The second Book of Geometry. Of a Line. 1. A Magnitude is either a Line or a Lineate. THe Common affections of a magnitude are hitherto declared: The Species or kinds do follow: for other than this division our author could not then meet withal. 2. A Line is a Magnitude only long. 3. The bound of a line is a point. 4. A Line is either Right or Crooked. This division is taken out of the 4 d i. of Euclid, where rectitude or straightness is attributed to a line, as if from it both surfaces and bodies were to have it. And even so the rectitude of a solid figure, hereafter shall be understood by a right line perpendicular from the top unto the centre of the base. Wherefore rectitude is proper unto a line. And therefore also obliquity or crookedness, from whence a surface is judged to be right or oblique, and a body right or oblique. 5. A right line is that which lieth equally between his own bounds: A crooked line lieth chose. 4. d. i. Therefore 6. A right line is the shortest between the same bounds. Linea recta, a strait or right line is that, as Plato defineth it, whose middle points do hinder us from seeing both the extremes at once; As in the eclipse of the Sun, if a right line should be drawn from the Sun, by the Moon, unto our eye, the body of the Moon being in the midst, would hinder our sight, and would take away the sight of the Sun from u●● which is taken from the Optics, in which we are taught, that we see by strait beams or rays. Therefore to lie equally between the bounds, that is by an equal distance: to be the shortest between the same bounds; And that the midst doth hinder the sight of the extremes, is all one. 7. A crooked line is touched of a right or crooked line, when they both do so meet, that being continued or drawn out farther they do not cut one another. Tactus, Touching is proper to a crooked line, compared either with a right line or crooked, as is manifest out of the 2. and 3. d 3. A right line is said to touch a circle, which touching the circle and drawn out farther, doth not cut the circle, 2 d 3. as here a e, the right line toucheth the periphery i o u. And a e. doth touch the helix or spiral. Circles are said to touch one another, when touching they do not cut one another, 3. d 3. as here the periphery doth a e i. doth touch the periphery o u y. Therefore 8. Touching is but in one point only. è 13. p 3. This Consectary is immediately conceived out of the definition; for otherwise it were a cutting, not touching. So Aristotle in his Mechanickes saith; That a round is easiliest moved and most swift; Because it is least touched of the plain underneath it. 9 A crooked line is either a Periphery or an Helix. This also is such a division, as our Author could then hit on. 10. A Periphery is a crooked line, which is equally distant from the midst of the space comprehended. Therefore 11. A Periphery is made by the turning about of a line, the one end thereof standing still, and the other drawing the line. Now the line that is turned about, may in a plain, be either a right line or a crooked line: In a spherical it is only a crooked line; But in a conical or Cylindraceall it may be a right line, as is the side of a Cone and Cylinder. Therefore in the conversion or turning about of a line making a periphery, there is considered only the distance; yea two points, one in the centre, the other in the top, which therefore Aristotle nameth Rotundi principia, the principles or beginnings of a round. 12. An Helix is a crooked line which is unequally distant from the midst of the space, howsoever enclosed. 13. Lines are right one unto another, whereof the one falling upon the other, lieth equally: chose they are oblique. è 10. dj. Hitherto straightness and crookedness have been the affections of one sole line only: The affections of two lines compared one with another are Perpendiculum, Perpendicularity and Parallelismus, Parallel equality; Which affections are common both to right and crooked lines. Perpendicularity is first generally defined thus: Therefore, 14. If a right line be perpendicular unto a right line, it is from the same bound, and on the same side, one only. ê 13. p. xj. 15. Parallel lines they are, which are every where equally distant. è 35. dj. Parallelismus, Parallell-equality doth now follow: And this also is common to crooked lines and right lines: As here thou seest in these examples following. Parallell-equality is derived from perpendicularity, and is of near affinity to it. Therefore Posidonius did define it by a common perpendicle or plumline: yea and in deed our definition intimateth as much. Parallell-equality of bodies is no where mentioned in Euclides● Elements: and yet they may also be parallels, and are often used in the Optickes, Mechanickes, Painting and Architecture. Therefore, 16. Lines which are parallel to one and the same line, are also parallel one to another. The third Book of Geometry. Of an Angle. 1. A lineate is a Magnitude more than long. A New form of doctrine hath forced our Author to use oft times new words, especially in dividing, that the logical laws and rules of more perfect division by a dichotomy, that is into two kinds, might be held and observed. Therefore a Magnitude was divided into two kinds, to wit into a Line and a Lineate: And a Lineate is made the genus of a surface and a Body. Hitherto a Line, which of all bignesses is the first and most simple, hath been described: Now followeth a Lineate, the other kind of magnitude opposed as you see to a line, followeth next in order. Lineatum therefore a Lineate, or Lineamentum, a Lineament, (as by the authority of our Author himself, the learned Bernhard Salignacus, who was his Scholar, hath corrected it) is that Magnitude in which there are lines: Or which is made of lines, or as our Author here, which is more than long: Therefore lines may be drawn in a surface, which is the proper soil or plots of lines: They may also be drawn in a body, as the Diameter in a Prisma: the axis in a sphere; and generally all lines falling from aloft: And therefore Proclus maketh some plain, other solid lines. So conical lines, as the Ellipsis, Hyperbole, and Parabole, are called solid lines because they do arise from the cutting of a body 2. To a Lineate belongeth an Angle and a Figure. The common affections of a Magnitude were to be bounded, cut, jointly measured, and ascribed: Then of a line to be right, crooked, touched, turned about, and wreathed: All which are in a lineate by means of a line. Now the common affections of a Lineate are to be Angled and Figured. And surely an Angle and a Figure in all Geometrical businesses do fill almost both sides of the leaf. And therefore both of them are diligently to be considered. 3. An Angle is a lineate in the common section of the bounds. So Angulus superficiarius, a superficial Angle, is a surface consisting in the common section of two lines: So angulus solidus, a solid angle, in the common section of three surfaces at the least. 4. The shanks of an angle are the bounds compreding the angle. Scèle or Crura, the Shankes, Legs, H. are the bounds insisting or standing upon the base of the angle, which in the Isosceles only or Equicrural triangle are so named of Euclid, otherwise he nameth them La●era, sides. So in the examples aforesaid, e a. and e i. are the shanks of the superficiary angle e● And so are the three surfaces a e i. i e e o. and a e o. the shanks of the said angle o. Therefore the shanks making the angle are either Lines or Surfaces: And the lineates form or made into Angles, are either Surfaces or Bodies. 5. Angel's homogeneal, are angles of the same kind, both in respect of their shanks, as also in the manner of meeting of the same: [Heterogeneal, are those which differ one from another in one, or both these conditions.] 6. Angels congruall in shanks are equal. This is drawn out of the 10. e i. For if twice two shanks do agree, they are not four, but two shanks, neither are they two equal angles, but one angle. And this is that which Proclus speaketh of, at the 4. p i. when he saith, that a right lined angle is equal to a right lined angle, when one of the shanks of the one put upon one of the shanks of the other, the other two do agree: when that other shank fall without, the angle of the out-falling shank is the greater: when it falleth within, it is lesser: For there is comprehendeth; here it is comprehended. The same Lunular also may be equal to an obtusangle and Acutangle, as the same argument will demonstrate. Therefore, 7. If an angle being equicrural to an other angle, be also equal to it in base, it is equal: And if an angle having equal shanks with another, be equal to it in the angle, it is also equal to it in base. è 8. & 4. pj. For such angles shall be congruall or agreeable in shanks, and also congruall in bases. Angulus isosceles, or Angulus oequicrurus, is a triangle having equal shanks unto another. 8. And if an angle equal in base to another, be also equal to it in shanks, it is equal to it. And 9 If an angle equicrural to another angle, be greater than it in base, it is greater: And if it be greater, it is greater in base: è 52 & 24. pj. And 10. If an angle equal in base, be less in the inner shanks, it is greater. Or as the learned Master T. Hood doth paraphrastically translate it. If being equal in the base, it be lesser in the feet (the feet being contained within the feet of the other angle) it is the greater angle. [That is, if one angle enscribed within another angle, be equal in base, the angle of the inscribed shall be greater than the angle of the circumscribed.] Therefore, 11. If unto the shanks of an angle given, homogeneal shanks, from a point assigned, be made equal upon an equal base, they shall comprehend an angle equal to the angle given. è 23. p i. & 26. p xj. [This consectary teacheth how unto a point given, to make an angle equal to an Angle given. To the effecting and doing of each three things are required; First, that the shanks be homogeneal, that is in each place, either strait or crooked: Secondly, that the shanks be made equal, that is of like or equal bigness: Thirdly, that the bases be equal: which three conditions if they do meet, it must needs be that both the angles shall be equal: but if one of them be wanting, of necessity again they must be unequal.] This shall hereafter be declared and made plain by many and sundry practices: and therefore here we bring no example of it. 12. An angle is either right or oblique. Thus much of the Affections of an angle; the division into his kinds followeth. An angle is either Right or Oblique: as afore, at the 4 e ij. a line was right or strait, and oblique or crooked. 13. A right angle is an angle whose shanks are right (that is perpendicular) one unto another: An Oblique angle is contrary to this. Therefore, 14. All straight-shanked right angles are equal. [That is, they are alike, and agreeable, or they do fill the same place; as here are a i o. and e i o. And yet again on the contrary: All strait shanked equal angles, are not rightangles.] Angel's therefore homogeneal and recticrurall, that is whose shanks are right, as are right lines, as plain surfaces (For let us so take the word) are equal right angles. So are the above written rectilineall right angles equal: so are plain solid right angles, as in a cube, equal. The axiom may therefore generally be spoken of solid angles, so they be recticruralls; Because all semicircular right angles are not equal to all semicircular right angles: As here, when the diameter is continued it is perpendicular, and maketh twice two angles, within and without, the outer equal between themselves, and inner equal between themselves: But the outer unequal to the inner: And the angle of a greater semicircle is greater, than the angle of a lesser. Neither is this affection any way reciprocal, That all equal angles should be right angles. For oblique angles may be equal between themselves: And an oblique angle may be made equal to a right angle, as a Lunular to a rectilineall right angle, as was manifest, at the 6 e. The definition of an oblique is understood by the obliquity of the shanks: whereupon also it appeareth; That an oblique angle is unequal to an homogeneal right angle: Neither indeed may oblique angles be made equal by any law or rule: Because obliquity may infinitely be both increased and diminished. 15. An oblique angle is either Obtuse or Acute. One difference of Obliquity we had before at the 9 e ij. in a line, to wit of a periphery and an helix; Here there is another dichotomy of it into obtuse and acute: which difference is proper to angles, from whence it is translated or conferred upon other things and metaphorically used, as Ingenium-obtusum, acutum; A dull, and quick wit, and such like. 16. An obtuseangle is an oblique angle greater than a right angle. 11. dj. 17. An acutangle is an oblique angle lesser than a right angle. 12. dj. The fourth Book, which is of a Figure. 1. A figure is a lineate bounded on all parts. SO the triangle a e i. is a figure; Because it is a plain bounded on all parts with three sides. So a circle is a figure: Because it is a plain every way bounded with one periphery. 2. The centre is the middle point in a figure. In some part of a figure the Centre, Perimeter, Radius, Diameter and Altitude are to be considered. The Centre therefore is a point in the midst of the figure; so in the triangle, quadrate, and circle, the centre is, a e i. Centrum gravitatis, the centre of weight, in every plain magnitude is said to be that, by the which it is handled or held up parallel to the horizon: Or it is that point whereby the weight being suspended doth rest, when it is carried. Therefore if any plate should in all places be alike heavy, the centre of magnitude and weight would be one and the same. 3. The perimeter is the compass of the figure. 4. The Radius is a right line drawn from the centre to the perimeter. 5. The Diameter is a right line inscribed within the figure by his centre. Therefore, 6. The diameters in the same figure are infinite. Although of an infinite number of unequal lines that be only the diameter, which passeth by or through the centre notwithstanding by the centre there may be divers and sundry. In a circle the thing is most apparent: as in the Astrolabe the index may be put up and down by all the points of the periphery. So in a spear and all rounds the thing is more easy to be conceived, where the diameters are equal: yet notwithstanding in other figures the thing is the same. Because the diameter is a right line inscribed by the centre, whether from corner to corner, or side to side, the matter skilleth not. Therefore that there are in the same figure infinite diameters, it issueth out of the definition of a diameter. And 7. The centre of the figure is in the diameter. This consectary, saith the learned Rod. Snellius, is as it were a kind of invention of the centre. For where the diameters do meet and cut one another, there must the centre needs be. The cause of this is for that in every figure there is but one centre only: And all the diameters, as before was said, must needs pass by that centre. And 8. It is in the meeting of the diameters. 9 The Altitude is a perpendicular line falling from the top of the figure to the base. 10. An ordinate figure, is a figure whose bounds are equal and angles equal. In plains the Equilater triangle is only an ordinate figure, the rest are all inordinate: In quadrangles, the Quadrate is ordinate, all other of that sort are inordinate: In every sort of Multangles, or many cornered figures one may be an ordinate. In crooked lined figures the Circle is ordinate, because it is contained with equal bounds, (one bound always equal to itself being taken for infinite many,) because it is equiangled, seeing (although in deed there be in it no angle) the inclination notwithstanding is every where alike and equal, and as it were the angle of the perphery be always alike unto itself: whereupon of Plato and Plutarch a circle is said to be Polygonia, a multangle; and of Aristotle Holegonia, a totangle, nothing else but one whole angle. In mingled-lined figures there is nothing that is ordinate: In solid bodies, and pyramids the Tetrahedrum is ordinate: Of Prismas, the Cube: of Polyhedrum's, three only are ordinate, the octahedrum, the Dodecahedrum, and the Icosahedrum. In oblique-lined bodies, the sphere is concluded to be ordinate, by the same argument that a circle was made to be ordinate. 11. A prime or first figure, is a figure which cannot be divided into any other figures more simple than itself. So in plains the triangle is a prime figure, because it cannot be divided into any other more simple figure although it may be cut many ways: And in solids, the Pyramid is a first figure: Because it cannot be divided into a more simple solid figure, although it may be divided into an infinite sort of other figures: Of the Triangle all plains are made; as of a Pyramid all bodies or solids are compounded● such are a e i. and a e i o. 12. A rational figure is that which is comprehended of a base and height rational between themselves. So Euclid, at the 1. d. ij. saith, that a rightangled parallelogramme is comprehended of two right lines perpendicular one to another, videlicet one multiplied by the other. For Geometrical comprehension is sometimes as it were in numbers a multiplication: Therefore if ye shall grant the base and height to be rationals between themselves, that their reason I mean may be expressed by a number of the assigned measure, than the numbers of their ●ides being multiplied one by another, the bigness of the figure shall be expressed. Therefore a Rational figure is made by the multiplying of two rational sides between themselves. Therefore, 13. The number of a rational figure, is called a Figurate number: And the numbers of which it is made, the Sides of the figurate. As if a Right angled parallelogramme be comprehended of the base four, and the height three, the Rational made shall be 12. which we here call the figurate: and 4. and 3. of which it was made, we name sides. 14. Isoperimetrall figures, are figures of equal perimeter. This is nothing else but an interpretation of the Greek word; So a triangle of 16. foot about, is a isoperimeter to a triangle 16. foot about, to a quadrate 16. foot about, and to a circle 16. foot about. 15. Of isoperimetralls homogenealls that which is most ordinate, is greatest: Of ordinate isoperimetralls heterogenealls, that is greatest, which hath most bounds. So an equilater triangle shall be greater than an isoperimeter inequilater triangle; and an equicrural, greater than an unequicrurall: so in quadrangles, the quadrate is greater than that which is not a quadrate: so an oblong more ordinate, is greater than an oblong less ordinate. So of those figures which are heterogeneal ordinates', the quadrate is greater than the Triangle: And the Circle, than the Quadrate. 16. If prime figures be of equal height, they are in reason one unto another, as their bases are: And chose. Therefore, 17. If prime figures of equal height have also equal bases, they are equal. [The reason is, because then those two figures compared, have equal sides, which do make them equal between themselves; For the parts of the one applied or laid unto the parts of the other, do fill an equal place, as was taught at the 10. e. i. Sn.] So Triangles, so Parallelogrammes, and so other figures proposed are equalled upon an equal base. 18. If prime figures be reciprocal in base and height, they are equal: And chose. 19 Like figures are equiangled figures, and proportional in the shanks of the equal angles. First like figures are defined, then are they compared one with another, similitude of figures is not only of prime figures, and of such as are compounded of prime figures, but generally of all other whatsoever. This similitude consisteth in two things, to wit in the equality of their angles, and proportion of their shankes● Therefore, 20. Like figures have answerable bounds subtended against their equal angles: and equal if they themselves be equal. Or thus, They have their terms subtended to the equal angles correspondently proportional: And equal if the figures themselves be equal; H. This is a consectary out of the former definition. And 21. Like figures are situate alike, when the proportional bounds do answer one another in like situation. The second consectary is of situation and place. And this like situation is then said to be when the upper parts of the one figure do agree with the upper parts of the other, the lower, with the lower, and so the other differences of places. Sn. And 22. Those figures that are like unto the same, are like between themselves. This third consectary is manifest out of the definition of like figures. For the similitude of two figures doth conclude both the same equality in angles and proportion of sides between themselves. And 23. If unto the parts of a figure given, like parts and alike situate, be placed upon a bound given, a like figure and likely situate unto the figure given, shall be made accordingly. This fourth consectary teacheth out of the said definition, the fabric and manner of making of a figure alike and likely situate unto a figure given. Sn. 24. Like figures have a reason of their homologallor correspondent sides equally manifold unto their dimensions: and a mean proportional less by one. Thus far of the first part of this element: The second, that like figurs have a mean, proportional less by one, then are their dimensions, shall be declared by few words. For plains having but two dimensions, have but one mean proportional, solids having three dimensions, have two mean proportionals. The ca●se is only Arithmetical, as afore. For where the bounds are but 4. as they are in two plains, there can be found no more but one mean proportional, as in the former example of 8. and 18. where the homologal or correspondent sides are 2. 3. and 4. 6. Therefore, Again by the same ru●e, where the bounds are 6. as they are in two solids, there may be found no more but two mean proportionals: as in the former solids 30. and 240. where the homologal or correspondent sides are 2. 4. 3. 6. 5. 10. Therefore, Therefore,) 25. If right lines be continually proportional, more by one then are the dimensions of like figures likelily situate unto the first and second, it shall be as the first right line is unto the last, so the first figure shall be unto the second: And chose. Out of the similitude of figures two consectaries do arise, in part only, as is their axiom, rational and expressable by numbers. If three right lines be continually proportional, it shall be as the first is unto the third: So the rectineall figure made upon the first, shall be unto the rectilineall figure made upon the second, alike and likelily situate. This may in some part be conceived and understood by numbers. As for example, Let the lines given, be 2. foot, 4. foot. and 8 foot. And upon the first and second, let there be made like figures, of 6. foot and 24. foot; So I mean, that 2. and 4. be the bases of them. Here as 2. the first line, is unto 8. the third line: So is 6. the first figure, unto 24. the second figure, as here thou seest. Again, let four lines continually proportional, be 1. 2. 4. 8. And let there be two like solids made upon the first and second: upon the first, of the sides 1. 3. and 2. lee it be 6. Upon the second, of the sides 2. 6. and 4. let it be 48. As the first right line 1. is unto the fourth 8. So is the figure 6. unto the second 48. as is manifest by division. The examples are thus. And 26. If four right lines be proportional between themselves: Like figures likelily situate upon them, shall be also proportional between themselves: And chose, out of the 22. puj. and 37. pxj. The proportion may also here in part be expressed by numbers: And yet a continual is not required, as it was in the former. In Plains let the first example be, as followeth. The cause of proportional figures, for that twice two figures have the same reason doubled. In Solids let this be the second example. And yet here the figures are not proportional unto the right lines, as before figures of equal height were unto their basest but they themselves are proportional one to another. And yet are they not proportional in the same kind of proportion. The cause also is here the same, that was before: To wit, because twice two figures have the same reason trebled. 27. Figures filling a place, are those which being any way set about the same point, do leave no void room. This was the definition of the ancient Geometers, as appeareth out of Simplicius, in his commentaries upon the 8. chapter of Aristotle's iij. book of Heaven: which kind of figures Aristotle in the same place deemeth to be only ordinate, and yet not all of that kind● But only three among the Plains, to wit a Triangle, a Quadrate, and a Sexangle: amongst Solids, two; the Pyramid, and the Cube. But if the filling of a place be judged by right angles, 4. in a Plain, and 8. in a Solid, the Oblong of plains, and the Octahedrum of Solids shall (as shall appear in their places) fill a place; And yet is not this Geometry of Aristotle accurate enough. But right angles do determine this sentence, and so doth Euclid out of the angles demonstrate, That there are only five ordinate solids; And so doth Potamon the Geometer, as Simplicus testifieth, demonstrate the question of figure's filling a place. Lastly, if figures, by laying of their corners together, do make in a Plain 4. right angles, or in a Solid 8. they do fill a place. Of this problem the ancient geometers have written, as we heard even now: And of the latter writers, Regiomontanus is said to have written accurately; And of this argument Maucolycus hath promised a treatise, neither of which as yet it hath been our good hap to see. Neither of these are figures of this nature, as in their due places shall be proved and demonstrated. 28. A round figure is that, all whose rays are equal. Rotundum, a Roundle, let it be here used for Rotunda figura, a round figure. And in deed Thomas Finkius or Finche, as we would call him, a learned Dane, sequestering this argument from the rest of the body of Geometry, hath entitled that his work De Geometria rotundi, Of the Geometry of the Round or roundle. 29. The diameters of a roundle are cut in two by equal rays. The reason is, because the halves of the diameters, are the rays. Or because the diameter is nothing else but a doubled ray: Therefore if thou shalt cut off from the diameter so much, as is the radius or ray, it followeth that so much shall still remain, as thou hast cut of, to wit one ray, which is the other half of the diameter. Sn. And here observe, That Bisecare, doth here, and in other places following, signify to cut a thing into two equal parts or portions● And so Bisegmentum, to be one such portion● And Bisectio, such a like cutting or division. 30. Rounds of equal diameters are equal. Out of the 1. d. i●●. Circles and Spheres are equal, which have equal diameters. For the rays, which do measure the space between the Centre and Perimeter, are equal, of which, being doubled, the Diameter doth consist. Sn. The fifth Book, of Ramus his Geometry, which is of Lines and Angles in a plain Surface. 1. A lineate is either a Surface or a Body. LIneatum, (or Lineamentum) a magnitude made of lines, as was defined at 1. e. iij. is here divided into two kinds: which is easily conceived out of the said definition there, in which a line is excluded, and a Surface & a body are comprehended. And from hence arose the division of the art Metriall into Geometry, of a surface, and Stereometry, of a body, after which manner Plato in his seven. book of his Commonwealth, and Aristotle in the 7. chapter of the first book of his Posteriorums, do distinguish between Geometry and Stereometry: And yet the name of Geometry is used to signify the whole art of measuring in general. 2. A Surface is a lineate only broad. 5. dj. Epiphania, the Greek word, which importeth only the outer appearance of a thing, is here more significant, because of a Magnitude there is nothing visible or to be seen, but the surface. 3. The bound of a surface is a line. 6. dj. The matter in Plains is manifest. For a three cornered surface is bounded with 3. lines: A four cornered surface, with four li●es, and so forth: A Circle is bounded with one line. But in a Sphearicall surface the matter is not so plain: For it being whole, seemeth not to be bounded with a line. Yet if the manner of making of a Sphearicall surface, by the conversion or turning about of a semiperiphery, the beginning of it, as also the end, shallbe a line, to wit a semiperiphery: And as a point doth not only actu, or indeed bound and end a line: But is potentia, or in power, the midst of it: So also a line boundeth a Surface actu, and an innumerable company of lines may be taken or supposed to be throughout the whole surface. A Surface therefore is made by the motion of a line, as a Line was made by the motion of a point. 4. A surface is either Plain or Bowed. The difference of a Surface, doth answer to the difference of a Line● in straightness and obliquity or crookedness. Obliquum, oblique, there signified crooked; Not righ● or strait: Here, uneven or bowed, either upward or downward. Sn. 5. A plain surface is a surface, which lieth ●qually between his bounds, out of the 7. dj. Planum, a Plain, is taken and used for a plain surface: as before Rotundum, a Round, was used for a round figure. Therefore, 6. From a point unto a point we may, in a plain surface, draw a right line. 1 and 2. post. i. Three things are from the former ground begged: The first is of a Right line. A right line and a periphery were in the ij. book defined: But the fabric or making of them both, is here said to be properly in a plain. Now the Geometrical instrument for the drawing of a right plain is called Amussis, & by Petolemey, in the 2. chapter of his first book of his Music, Regula, a ruler, such as here thou seest. And from a point unto a point is this justly demanded to be done, not unto points; For neither do all points fall in a right line: But many do fall out to be in a crooked line. And in a Sphere, a Cone & Cylinder● a Ruler may be applied, but it must be a sphearicall, conical, or Cylindraceall. But by the example of a right line doth Vitellio, 2 p i. demand that between two lines a surface may be extended: And so may it seem in the Elements, of many figures both plain and solids, by Euclid to be demanded; That a figure may be described, at the 7. and 8. e ij. Item that a figure may be made up, at the 8. 14. 16. 23.28 p. uj. which are of Plains. Item at the 25. 31. 33. 34. 36. 49. p.xj. which are of Solids. Yet notwithstanding a plain surface, and a plain body do measure their rectitude by a right line, so that jus postulandi, this right of begging to have a thing granted may seem primarily to be in a right plain line. Now the Continuation of a right line is nothing else, but the drawing out farther of a line now drawn, and that from a point unto a point, as we may continue the right line a e. unto i. wherefore the first and second Petitions of Eu●lde do agree in one. And] 7. To set at a point assigned a Right line equal to another right line given: And from a greater, to cut off a part equal to a lesser. 2. and 3. pj. Therefore, 8. One right line, or two cutting one another, are in the same plain, out of the 1. and 2. p xj. One Right line may be the common section of two plains: yet all or the whole in the same plain is one: And all the whole is in the same other: And so the whole is the same plain. Two Right lines cutting one another, may be in two plains cutting one of another; But then a plain● may be drawn by them: Therefore both of them shall be in the same plain. And this plain is geometrically to be conceived: Because the same plain is not always made the ground whereupon one oblique line, or two cutting one another are drawn, when a periphery is in a sphearicall: Neither may all peripheries cutting one another be possibly in one plain. And 9 With a right line given to describe a periphery. Talus, the nephew of Daedalus by his sister, is said in the viij. book of Ovid's Metamorphosis, to have been the inventour of this instrument: For there he thus writeth of him and this matter:— Et ex uno duo ferrea brachia nodo: junxit, ut aequali spatio distantibus ipsis: Altera pars staret, pars altera duce●et orbem. Therefore 10. The rai●s of the same, or of an equal periphery, are equal. The reason is, because the same right line is every where converted or turned about. But here by the Ray of the periphery, must be understood the Ray the figure contained within the periphery. 11. If two equal peripheries, from the ends of equal shanks of an assigned rectilineall angle, do meet before it, a right line drawn from the meeting of them unto the top or point of the angle, shall cut it into two equal parts. 9 pj. Hitherto we have spoken of plain lines: Their affection followeth, and first in the Bisection or dividing of an Angle into two equal parts. 12. If two equal peripheries from the ends of a right line given, do meet on each side of the same, a right line drawn from those meetings, shall divide the right line given into two equal parts. 10. pj. 13. If a right line do stand perpendicular upon another right line, it maketh on each side right angles: And contrary wise. The ruler, for the making of strait lines on a plain, was the first Geometrical instrument: The Compasses, for the describing of a Circle, was the second: The Norma or Square for the true erecting of a right line in the same plain upon another right line, and then of a surface and body, upon a surface or body, is the third. The figure therefore is thus. Therefore 14. If a right line do stand upon a right line, it maketh the angles on each side equal to two right angles: and chose out of the 13. and 14. pj. And 15. If two right lines do cut one another, they do make the angles at the top equal and all equal to four right angles. 15. pj. And 16. If two right lines cut with one right line, do make the inner angles on the same side greater than two right angles, those on the other side against them shall be lesser than two right angles. 17. If from ●●oint assigned of an infinite right line given, two equal parts be on each side cut off: and then from the points of those sections two equal circles do meet, a right line drawn from their meeting unto the point assigned, shall be perpendicular unto the line given. 11. pj. 18. If a part of an infinite right line, be by a periphery from a point given without, cut off a right line from the said point, cutting in two the said part, shall be perpendicular upon the line given. 12. pj. 19 If two right lines drawn at length in the same plain do never meet, they are parallelly. è 35. dj. Therefore 20. If an infinite right line do cut one of the infinite right parallel lines, it shall also cut the other. As in the same example u y. cutting a e. it shall also cu● i o. Otherwise, if it should not cut it, it should be parallel unto it, by the 18 e. And that against the grant. 21. If right lines cut with a right line be pararellells, they do make the inner angles on the same side equal to two right angles: And also the alterne angles equal between themselves: And the outer, to the inner opposite to it: And chose, 29,28,27. p 1. The cause of this threefold propriety is from the perpendicular or plumbline, which falling upon the parallels breedeth and discovereth all this variety: As here they are right angles which are the inner on the same part or side: Item, the alterne angles: Item the inner and the outer: And therefore they are equal, both, I mean, the two inner to two right angles: and the alterne angles between themselves: And the outer to the inner opposite to it. If so be that the cutting line be oblique, that is, fall not upon them plumb or perpendicularly, the same shall on the contrary befall the parallels. For by that same obliquation or slanting, the right lines remaining and the angles unaltered, in like manner both one of the inner, to wit, e u y, is made obtuse, the other, to wi●, u y o, is made acute: And the alterne angles are made acute and obtuse: As also the outer and inner opposite are likewise made acute and obtuse. The same impossibility shall be concluded, if they shall be said, to be lesser than two right angles● The second and third parts may be concluded out of the first. The second is thus: Twice two angles are equal to two right angles o y u, and e u y, by the former part: Item, a u y, and e u y, by the 14 e. Therefore they are equal between themselves. Now from the equal, Take away e u y, the common angle, And the remainders, the alterne angles, at u, and y shall be least equal. The third is thus; The angles e u y, and oh y s, are equal to the same u y i, by the second propriety, and by the 15 e. Therefore they are equal between themselves. If they be oblique angles, as here, the lines one slanting or liquely crossing one another, the angles on one side will grow less, on the other side greater. Therefore they would not be equal to two right angles, against the grant. From hence the second and third parts may be concluded. The second is thus: The alterne angles at u and y, are equal to the foresaid inner angles, by the 14 e: Because both of them are equal to the two right angles: And so by the first part the second is concluded. The third is therefore by the second demonstrated, because the outer oh y s, is equal to the vertical or opposite angle at the top, by the 15 e. Therefore seeing the outer and inner opposite are equal, the alterne also are equal. Wherefore as Parallelismus, parallell-equality argueth a threefold equality of angels: So the threefold equality of angles doth argue the same parallel-equality. Therefore, 22. If right lines knit together with a right line, do make the inner angles on the same side lesser than two right Angles, they being on that side drawn out at length, will meet. And 23. A right line knitting together parallel right lines, is in the same plain with them. 7 p xj. And 24. If a right line from a point given do with a right line given make an angle, the other shank of the angle equalled and alterne to the angle made, shall be parallel unto the assigned right line. 31 pj. An angle, I confess, may be made equal by the first propriety: And so indeed commonly the Architects and Carpenters do make it, by erecting of a perpendicular. It may also again in like manner be made by the outer angle: Any man may at his pleasure use which he shall think good; But that here taught we take to be the best. And 25. The angles of shanks alternly parallel, are equal. Or Thus, The angles whose altenate feet are parallels, are equal. H. And 26 If parallels do bound parallels, the opposite lines are equal è 34 p.j. Or thus: If parallels do enclose parallels, the opposite parallels are equal. H. And 27. If right lines do jointly bound on the same side equal and parallel lines, they are also equal and parallel. On the same part or side it is said, lest any man might understand right lines knit together by opposite bounds as here. 28. If right lines be cut jointly by many parallel right lines, the segments between those lines shall be proportional one to another, out of the 2 p uj and 17 p x i. Thus much of the Perpendicle, and parallel equality of plain right lines: Their Proportion is the last thing to be considered of them. If the lines cut be not parallels, but do lean one toward another, the portions cut or intercepted between them will not be equal, yet shall they be proportional one to another. And look how much greater the line thus cut is: so much greater shall the intersegments or portions intercepted be. And chose, Look how much less: so much lesser shall they be. The third parallel in the top is not expressed, yet must it be understood. This element is very fruitful: For from hence do arise and issue, First the manner of cutting a line according to any rate or proportion assigned: And then the invention or way to find out both the third and fourth proportionals. 29. If a right line making an angle with another right line, be cut according to any reason [or proportion] assigned, parallels drawn from the ends of the segments, unto the end of the said right line given and unto some contingent point in the same, shall cut the line given according to the reason given. Schoner hath altered this Consectary, and delivereth it thus: If a right making an angle with a right line given, and 〈◊〉 it unto it with a base, be cut according to any rate assigned, a parallel to the base from the ends of the segments, shall cut the line given according to the rate assigned. 9 and 10 p v i. Punctum contingens, A contingent point, that is falling or lighting in some place at all adventurs, not given or assigned This is a marvellous general consectary, serving indifferently for any manner of section of a right line, whether it be to be cut into two parts, or three parts, or into as many patts, as you shall think good, or generally after what manner of way soever thou shalt command or desire a line to be cut or divided. Now 〈◊〉 be cut into three partest 〈◊〉 which the first let it be the half of the second: And the second, the half of the third: And the conter minall or right line making an angle with the said assigned line, let it be cut one part a o: Then double this in o u: Lastly let u i be taken double to o u, and let the whole diagramme be made up with three parallels y● and os, The fourth parallel in the top, as a foresaid, shall be understood. Therefore that section which was made in the conterminall line, by the 28 e, shall be in the assigned line: Because the segments or portions intercepted are between the parallels. And 30. If two right lines given, making an angle, be continued, the first equally to the second, the second infinitely, parallels drawn from the ends of the first continuation, unto the beginning of the second, and some contingent point in the same, shall intercept between them the third proportional. 11. p v i. And 31. If of three right lines given, the first and the third making an angle be continued, the first equally to the second, and the third infinitely; parallels drawn from the ends of the first continuation, unto the beginning of the second, and some contingent point, the same shall intercept between them the fourth proportional. 12. p vi. Let the lines given be these: The first a e, the second e i, the third a o, and let the whole diagramme be made up according to the prescript of the consectary. Here by 28. e, as a e, is to e i so is a o, to o u. Thus far Ramus. Lazarus Schonerus, who, about some 25. years since, did revise and augment this work of our Author, hath not only altered the form of these two next precedent consectaries: but he hath also changed their order, and that which is here the second, is in his edition the third: and the third here, is in him the second. And to the former declaration of them, he addeth these words: From hence, having three lines given, is the invention of the fourth proportional; and out of that, having two lines given, ariseth the invention of the third proportional. 2 Having three right lines given, if the first and the third making an angle, and knit together with a base, be continued, the first equally to the second; the third infinitely; a parallel from the end of the second, unto the continuation of the third, shall intercept the fourth proportional. 12. puj. The Diagramme, and demonstration is the same with our 31. e or 3 c of Ramus. 3 If two right lines given making an angle, and knit together with a base, be continued, the first equally to the second, the second infinitely; a parallel to the base from the end of the first continuation unto the second, shall intercept the third proportional. 11. p v i. The Diagramme here also, and demonstration is in all respects the same with our 30 e, or 2 c of Ramus. Thus far Ramus: And here by the judgement of the learned Finkius, two elements of Ptolomey are to be adjoined. 32 If two right lines cutting one another, be again cut with many parallels, the parallels are proportional unto their next segments. The same demonstation shall serve, if the lines do cross one another, or do vertically cut one another, as in the same diagramme appeareth. For if the assigned a i, and u s, do cut one another vertically in o, let them be cut with the parallels a u, and s i: the precedent fabric or figure being made up, it shall be by 28. e. as a u, is unto a o, the segment next unto it: so a y, that is, i s, shall be unto o ay, his next segment. The 28. e teacheth how to find out the third and fourth proportional: This affordeth us a means how to find out the continually mean proportional single or double. Therefore 33. If two right lines given be continued into one, a perpendicular from the point of continuation unto the angle of the squire, including the continued line with the continuation, is the mean proportional between the two right lines given. A squire (Norma, Gnomon, or Canon) is an instrument consisting of two shanks, including a right angle. Of this we heard before at the 13. e. By the means of this a mean proportional unto two lines given is easily found: whereupon it may also be called a Mesolabium, or Mesographus simplex, or single mean finder. And 34 If two assigned right lines joined together by their ends right anglewise, be continued vertically; a square falling with one of his shanks, and another to it parallel and movable upon the ends of the assigned, with the angles upon the continued lines, shall cut between them from the continued two means continually proportional to the assigned. The former consectary was of a single mesolabium; this is of a double, whose use in making of solids, to this or that bigness desired is notable. And thus we have the composition and use, both of the single and double Mesolabium. 35. If of four right lines, two do make an angle, the other reflected or turned back upon themselves, from the ends of these, do cut the former; the reason of the one unto his own ●egment, or of the segments between themselves, is made of the reason of the so jointly bounded, that the first of the makers be jointly bounded with the beginning of the antecedent made; the second of this consequent jointly bounded with the end; do end in the end of the consequent made. Let therefore the two right lines be ● e, and a i: and from the ends of these other two reflected, be i u, and e o, cutting themselves in y; and the two former in u, and o. The reason of the particular right lines made shall be as the draught following doth manifest. In which the antecedents of the makers are in the upper place: the consequents are set under neathe their own antecedents. The business is the same in the two other, whether you do cross the bounds or invert them. Here for demonstrations sake we crave no more, but that from the beginning of an antecedent made a parallel be drawn to the second consequent of the makers, unto one of the assigned infinitely continued: then the multiplied proportions shall be, The Antecedent, the Consequent; the Antecedent, the Consequent of the second of the makers; every way the reason or rate is of Equality. The Antecedent the Consequent of the first of the makers; the Parallel; the Antecedent of the second of the makers, by the 32. e. Therefore by multiplication of proportions, the reason of the Parallel, unto the Consequent of the second of the makers, that is, by the fabric or construction, and the 32. e. the reason of the Antecedent of the Product, unto the Consequent, is made of the reason, etc. after the manner above written. Again, I say, that the reason of e y, unto y o, is compounded of the reason of e u, unto u a, and of a i, unto i ●. Theon here draweth a parallel from o, unto u i. By the general fabric it may be drawn out of e, unto o i. Therefore the reason of e n, unto i o, that is of e y, unto y o, shall be made of the foresaid reasons. Of the segments of divers right lines● the Arabians have much under the name of The rule of six quantities. And the Theorems of Althin●us, concerning this matter, are in many men's hands. And Regiomontanus in his Algorithmus: and Maurolycus upon the 1 piij. of Menelaus, do make mention of them; but they contain nothing, which may not; by any man skilful in Arithmetic, be performed by the multiplication of proportions. For all those ways of theirs are no more but special examples of that kind of multiplication. Of Geometry, the sixth Book, of a Triangle. 1 Like plains have a double reason of their hom●logall sides, and one proportional mean, out of 20 p vi. and xj. and 18. p viij. OR thus; Like plains have the proportion of their corespondent proportional sides doubled, & one mean proportional: Hitherto we have spoken of plain lines and their affections: Plain figures and their kinds do follow in the next place. And first, there is premised a common corollary drawn out of the 24. e, iiij. because in plains there are but two dimensions. 2 A plain surface is either rectilineall or obliquelineall, [or rightlined, or crookedlined. H.] Straightness, and crookedness, was the difference of lines at the 4. e, i i. From thence is it here repeated and attributed to a surface, which is geometrically made of lines. That made of right lines, is rectileniall: that which is made of crooked lines, is Obliquilineall. 3. A rectilineall surface, is that which is comprehended of right lines. 4 A rightilineall doth make all his angles equal to right angles; the inner ones generally to pairs from two forward: the outer always to four. Or thus: A right lined plain maketh his angles equal unto right angles: Namely the inward angles generally, are equal unto the even numbers from two forward, but the outward angles are equal but to 4. right angles. H. 5 A rectilineall is either a Triangle or a Triangulate. As before of a line was made a lineate: so here in like manner of a triangle is made a triangulate. 6 A triangle is a rectilineall figure comprehended of three rightlines. 21. dj. Therefore 7 A triangle is the prime figure of rectilineals. A triangle or threesided figure is the prime or most simple figure of all rectilineals. For amongst rectilineall figures there is none of two sides: For two right lines cannot enclose a figure. What is meant by a prime figure, was taught at the 7. e. iiij. And 8 If an infinite right line do cut the angle of a triangle, it doth also cut the base of the same: Vitell. 29. t i. 9 Any two sides of a triangle are greater than the other. Let the triangle be a e i; I say, the side a i, is shorter, than the two sides a e, and e i, because by the 6. e ij, a right line is between the same bounds the shortest. Therefore 10 If of three right lines given, any two of them be greater than the other, and peripheries described upon the ends of the one, at the distances of the other two, shall meet, the rays from that meeting unto the said ends, shall make a triangle of the lines given. And 11 If two equal peripheries, from the ends of a right line given, and at his distance, do meet, li●es drawn from the meeting, unto the said ends, shall make an equilater triangle upon the line given. 1 p.j. 12 If a right line in a triangle be parallel to the base, it doth cut the shanks proportionally: And chose. 2 p v i. As here in the triangle a e i, let o u, be parallel to the base; and let a third parallel be understood to be in the top a; therefore, by the 28. e.u. the intersegments are proportional. The converse is forced out of the antecedent: because otherwise the whole should be less than the part. For if o u, be not parallel to the base ei, then yu, is: Here by the grant, and by the antecedent, seeing a o, oh e, a y, y e, are proportional: and the first a o, is lesser than a y, the third: o e, the second must be lesser than y e, the fourth, that is the whole than the part. 13 The three angles of a triangle, are equal to two right angles. 32. p i. Therefore 14. Any two angles of a triangle are less than two right angles. For if three angles be equal to two right angles, then are two lesser than two right angles. And 15 The one side of any triangle being continued or drawn out, the outer angle shall be equal to the two inner opposite angles. Therefore 16 The said outer angle is greater than either of the inner opposite angles. 16. p i. This is a consectary following necessarily upon the next former consectary. 17 If a triangle be equicrural, the angles at the base are equal: and chose, 5. and 6. p.j. Therefore 18 If the equal shanks of a triangle be continued or drawn out, the angles under the base shall be equal between themselves. And 19 If a triangle be an equilater, it is also an equiangle: And chose. And 20 The angle of an equilater triangle doth countervail two third parts of a right angle. Regio. 23. p i. For seeing that 3. angles are equal to 2. 1. must needs be equal to ⅔ And 21 Six equilater triangles do fill a place. 22 The greatest side of a triangle subtendeth the greatest angle; and the greatest angle is subtended of the greatest side. 19 and 18. p i. The converse is manifest by the same figure: As let the angle a e i, be greater than the angle a i e. Therefore by the same, 9 e iij. it is greater in base. For what is there spoken of angles in general, are here assumed specially of the angles in a triangle. 23 If a right line in a triangle, do cut the angle in two equal parts, it shall cut the base according to the reason of the shanks; and chose. 3. p v i. The mingled proportion of the sides and angles doth now remain to be handled in the last place. The Converse likewise is demonstrated in the same figure. For as e a, is to a i; so is e o, to o i: And so is e a, to a u, by the 12 e: therefore a i, and a u, are equal, Item the angles e a o, and o a i, are equal to the angles at u, and i, by the 21. e v● which are equal between themselves by the 17. e. Of Geometry, the seventh Book, Of the comparison of Triangles. 1 Equilater triangles are equiangles. 8. p.j. Thus forre of the Geometry, or affections and reason of one triangle; the comparison of two triangles one with another doth follow. And first of their rate or reason, out of their sides and angles: Whereupon triangles between themselves are said to be equilaters and equiangles. First out of the equality of the sides, is drawn also the equality of the angles. Triangles therefore are here jointly called equilaters, whose sides are severally equal, the first to the first, the second, to the second, the third to the third; although every several triangle be inequilaterall. Therefore the equality of the sides doth argue the equality of the angles, by the 7. e iij. As here. 2 If two triangles be equal in angles, either the two equicrurals, or two of equal either shank, or base of two angles, they are equilaters, 4. and 26. p i. Oh thus; If two triangles be equal in their angles, either in two angles contained under equal feet, or in two angles, whose side or base of both is equal, those angles are equilater. H. This element hath three parts, or it doth conclude two triangles to be equilaters three ways. 1. The first part is apparent thus: Let the two triangles be a e i, and o u y; because the equal angles at a, and o, are equicrural, therefore they are equal in base, by the 7. e iij. 3 The third part is thus forced: In the triangles a e i, and o u y, let the angles at e, and i, and u, and y, be equal, as afore: And a e. the base of the angle at i, be equal to o u, the base of angle at y: I say that the two triangles given are equilaters. For if the side e i, be greater than the side u y, let e s, be cut off equal to it, and draw the right line a s. Therefore by the antecedent, the two triangles, a e s, and o u y, equal in the angle of their equal shanks are equiangle; And the angle a s e, is equal to the angle oh y u, which is equal by the grant unto the angle a i e. Therefore a s e, is equal to a i e, the outer to the inner, contrary to the 15. e v. i. Therefore the base e i, is not unequal to the base u y, but equal. And therefore as above was said, the two triangles a e i, and o u y, equal in the angle of their equal shanks, are equilaters. 3. Triangles are equal in their three angles. And yet notwithstanding it is not therefore to be thought to be equiangle to it: For Triangles are then equiangles, when the several angles of the one, are equal to the several angles of the other: Not when all jointly are equal to all. Therefore 4. If two angles of two triangles given be equal, the other also are equal. All the three angles, are equal between themselves by the 3 e. Therefore if from equal you take away equal, those which shall remain shall be equal. 5. If a right triangle equicrural to a triangle be greater in base, it is greater in angle: And chose. 25. and 24. pj. 6. If a triangle placed upon the same base, with another triangle, be lesser in the inner shanks, it is greater in the angle of the shanks. This is a consectary drawn also out of the 10 e iij. As here in the triangle a e i, and a o ay, within it and upon the same base. Or thus: If a triangle placed upon the same ba●e with another triangle, be less than the other triangle, in regard of his feet, (those feet being contained within the feet of the other triangle) in regard of the angle contained under those feet, it is greater: H. 7. Triangles of equal height, are one to another as their bases are one to another. Thus far of the Reason or rate of triangles: The proportion of triangles doth follow; And first of a right line with the bases. It is a consectary out of the 16 e iiij. Therefore 8. Upon an equal base, they are equal. 9 If a right line drawn from the top of a triangle, do cut the base into two equal parts, it doth also cut the triangle into two equal parts: and it is the diameter of the triangle. 10. If a right line be drawn from the top of a triangle, unto a point given in the base (so it be not in the midst of it) and a parallel be drawn from the midst of the base unto the side, a right line drawn from the top of the said parallel unto the said point, shall cut the triangle into two equal parts. 11 If equiangled triangles be reciprocal in the shanks of the equal angle, they are equal: And chose. 15. p. vi. Or thus, as the learned M●. Brigges hath conceived it: If two triangles, having one angle, are reciprocal, etc. The converse, is concluded by the same sorites, but by saying all backward. For u a unto a e is, as u a o is unto o a e, by the 7 e: And as e a i, by the grant: Because they are equal: And as i a is unto a o, by the same. Wherefore u a is unto a e, as i a is unto a o. 12 If two triangles be equiangles, they are proportional in Shankes: And chose: 4 and 5. p. vi. Therefore, 13. If a right line in a triangle be parallel to the base, it doth cut off from it a triangle equiangle to the ●hole● but less in base. 14. If two trangles be proportional in the shanks of the equal angle, they are equiangles: 6 p vi. 15 If triangles proportional in shanks, and al●ernly parallel, do make an angle between them, their bases are but one right line continued. 32 p. vi. Or thus: If being proportional in their feet, and alternately parallels, they make an angle in the midst between them, they have their bases continued in a right line: H. The cause is out of the 14 e v. For they shall make on each side, with the falling line a i, two angles equal to two right angles. 16 If two triangles have one angle equal, another proportional in shanks, the third homogeneal, they are equiangles. 7. p. v. i. Of Geometry the eight Book, of the divers kinds of Triangles. 1 A triangle is either right angled, or obliquangled. The division of a triangle, taken from the angles, out of their common differences, I mean, doth now follow. But here first a special division, and that of great moment, as hereafter shall be in quadrangles and prismes. 2 A right angled triangle is that which hath one right angle: An obliquangled is that which hath none. 27. d i. A right angled triangle in Geometry is of special use and force; and of the best Mathematicians it is called Magister matheseos, the master of the Mathematics. Therefore 3 If two perpendicular lines be knit together, they shall make a right angled triangle. 4 If the angle of a triangle at the base, be a right angle, a perpendicular from the top shall be the other shanke● [and chose Schon.] As is manifest in the same example. 5 If a right angled triangle be equicrural, each of the angles at the base is the hal●e of a right angle: And chose. Therefore 6 If one angle of a triangle be equal to the other two, it is a right angle [And chose Schon.] Because it is equal to the half of two right angles, by the 13. e u.j. And 7 If a right line from the top of a triangle cutting the base into too equal parts be equal to the bisegment, or half of the base, the angle at the top is a right angle: [And chose Schon.] 8 A perpendicular in a triangle from the right angle to the base, doth cut it into two triangles, like unto the whole and between themselves, 8. p v. i. [And chose Schon.] Therefore 9 The perpendicular is the mean proportional between the segments or portions of the base. As in the said example, as i o, is to o a: so is o a, to o e, because the shanks of equal angles are proportional, by the 8. e. From hence was Plato's Mesographus invented. And 10 Either of the shanks is proportional between the base, and the segment of the base next adjoining. For as e i, is unto i a, in the whole triangle, so is a i, to i o, in the greater. For so they are homologal sides, which do subtend equal angles, by the 23. e, iiij. Item, as i e, is to e a; in the whole triangle, so is a e, to e o, in the lesser triangle. Either of the shanks is proportional between the sum, and the difference of the base and the other shank. And chose. If one side be proportional between the sum and the difference of the others, the triangle given is a rectangle. M. H. Brigges. This is a consectary arising likewise out of the 4 e. of very great use. In the triangle e a d, the shank a d, 12. is the mean proportional between b d, 18. (the sum of the base a e, 13. and the shank e d, 5.) and 8. the difference of the said base and shank: For if thou shalt draw the right lines b a, and a c, the angle b ac, shall be by the 6. e, a rectangle; (because it is equal to the angles at b, and c, seeing that the triangles b e a, and e a c, are equicrural.) And by the 9 e, b d, d a, and d e, are continually proportional. If a quadrate of a number, given for the first shank, be divided of another, the half of the difference of the divisour, and quotient shall be the other shank, and the half of the sum shall be the base. Or thus, The side of divided number doubled, and the difference of the divisour and quotient, shall be the two shanks, and the sum of them shall be the base. Let the number given for the first shank be 4. And let 8. divide 16. the quadrate of 4. by 2. The half of 8— 2, that is 3. shall be the other shank: And the half of 8— 2, that is 5. shall be the base. Therefore If any one number shall divide the quadrate of another, the side of the divided, and the half of the difference of the divisour and the quotient, shall be the two shanks of a rectangled triangle, and the half of the sum of them shall be the base thereof. Let the two numbers given be 4. and 6. The square of 6. let it be 36. and the quotient of 36. by 4. be 9: And the side it 6. for the one shank. Now 9— 4. that is, 5. is the difference of the divisour and quotient, whose half 2. ½, is the other shank. And 9— 4. that is 13. is the sum the said devisour and quotient, whose half 6. ½, is the base. Again let 4. and 8. be given. The quadrate of 8. is 64. And the quiet of 64 is 16. and the side of 64. is 8. for the one shank. The half 16— 4. that is 6. is the other shank. And the half of 16— 4. that is 10, is the base. 11. If the base of a triangle do subtend a rightangle, the rectilineall fitted to it, shall be equal to the like rectilinealls in like manner fitted to the shanks thereof: And chose, out of the 31. p. v i. Or thus: If the base of a triangle do subtend a right angle, the right lined figure made upon the base, is equal to the right lined figures like, and in like manner situate upon the feet: H. The Converse is thus proved: Let the triangle be a e i: And let the perpendicular e o, be erected upon a e, equal to e i: And draw a right line from o to a: Here by the former, the rectilinealls situate at o e, and e a, that is by the construction, at a e, and i e, are equal to the rightilineall at a o, made alike and situate a like: And by the grant they are equal, to the rectilineall at a i, made a like and situated alike. Therefore seeing the like rectilineals at a o, and a i, are equal; they have by the 20 e iiij, their homologal sides equal: And the two triangles are equiliters: And by the 1 e seven, equiangles. But a e o, is a right angle, by the construction: And a e i, is proved to be equal to the same a e o: Therefore, by the 13 e v. a e i, also is a right angle. 12 An obliquangled triangle is either Obtusangled or Acutangled. The division of an obliquangled triangle is taken from the special differences of an oblique angle. For at the 15 e iij, we were taught that an oblique angle was either obtuse or acute: Therefore an obliquangled triangle is an obtuseangle, and an Acutangle. 13 An obtusangle is that triangle which hath one blunt corner, 28. di. There can be but one right angle in a triangle, by the 2 e. Therefore also in it there can be but one blunt angle. Therefore 14. If the obtuse or blunt angle be at the base of the triangle given, a perpendicular drawn from the top of the triangle, shall fall without the figure: And contrariwise. As here in a e i, the perpendicular i o, falleth without: This is manifest by the 4 e. And 15. If one angle of a triangle be greater than both the other two, it is an obtuse angle: And chose. This is plain by the 6 e. And 16. If a right line drawn from the top of the triangle cutting the base into two equal parts, be less than one of those halves, the angle at the top is a blunt-angle. And chose. 17. An acutangled triangle is that which hath all the angles acute. 29 dj. Therefore 18 A perpendicular drawn from the top falleth with out figure: And chose. As in a e i, the perpendicular a o falleth without as is plain by the 4 e. And 19 If any one angle of triangle be less than the other two, it is acute: And chose. As is manifest by the 6 e. And 20. If a right line drawn from the top of the triangle; cutting the base into two equal parts, be less than either of those portions, the angle at the top is an acute angle: And chose. The ninth Book, of P. Ramus Geometry, which intreateth of the measuring of right lines by like rightangled triangles. THe Geometry of like rightangled triangles, amongst many other uses that it hath, it doth especially afford us the geodaesy or measuring of right lines: And that mastery, which before (at the 2 e viij) attributed the right angled triangles, shall here be found to be a true mastery indeed. For it shall contain the geodesy of right lines; and afterward the geodesy of plains and solides, by the measuring of their sides, which are right lines. 1. For the measuring of right lines; we will use the Jacob's staff, which is a squire of unequal shanks. Radius, commonly called Baculus jacob, Jacob's staff, as if it had been long since invented and practised by that holy Patriarch, is a very ancient instrument, and of all other Geometrical instruments, commonly used, the best and fittest for this use. Archimedes in his book of the Number of the sand, seemeth to mention some such thing: And Hipparchus, with an instrument not much unlike this, boldly attempted an heinous matter in the sight of God, as Pliny thinketh, namely to deliver unto posterity the number of the stars, and to assign of fix them in their true places by the Norma, the squire or Jacob's staff. And indeed true it is that the Radius is not only used for the measuring of the earth and land: But especially for the defining or limiting of the stars in their places and order: And for the describing and setting out of all the regions and ways of the heavenly city. Yea and Virgil the famous Poet, in his 3 Ecloge, Ecquis fuit alter, Descripsit radio totum, qui gentibus orbem? and again afterward in the 6 of his Eneiades, hath noted both these uses. Coeliquè meatus. Describent radio & surgentia sidera dicent. Long after this the jews and Arabians, as Rabbi Levi; But in these latter days, the Germans especially, as Regiomontanus; Werner, Schoner, and Appian have graced it: But above all other the learned Gemma Phrisius in a several work of that argument only, hath illustrated and taught the use of it plainly and fully. The Jacob's staff therefore according to his own, and those Geometrical parts, shall here be described (The astronomical distribution we reserve to his time and place.) And that done, the use of it shall be showed in the measuring of lines. 2 The shanks of the staff are the Index and the Transome. 3 The Index is the double and one tenth part of the transome. Or thus: The Index is to the transversary double and 1/10 part thereof. H. As here thou seest. 4 The Transome is that which rideth upon the Index, and is to be slid higher or lower at pleasure. Or, The transversary is to be moved upon the Index, sometimes higher, sometimes lower: H. This proportion in defining and making of the shanks of the instrument is perpetually to be observed: as if the transome be 10. parts, the Index must be 21. If that be 189. this shall be 90. or if it be 2000 this shall be 4200. Neither doth it skill what the numbers be, so this be their proportion. More than this, That the greater the numbers be, that is the lesser that the divisions be, the better will it be in the use. And because the Index must bear, and the transome is to be borne; let the index be thicker, and the transome the thinner. But of what matter each part of the staff be made, whether of brass or wood it skilleth not, so it be firm, and will not cast or warp. Notwithstanding, the transome will more conveniently be moved up and down by brazen pipes, both by itself, and upon the Index higher or lower right angle wise, so touching one another, that the alterne mouth of the one may touch the side of the other. The third pipe is to be moved or slid up and down, from one end of the transome to the other; and therefore it may be called the Cursor. The fourth and fifth pipes, fixed and immovable, are set upon the ends of the transome, are unto the third and second of equal height with ●innes, to restrain when need is, the optic line, and as it were, with certain points to define it in the transome. The three first pipes may, as occasion shall require, be fastened or stayed with brazen screws. With these pipes therefore the transome may be made as great, as need shall require, as here thou seest. The fabric or manner of making the instrument hath hitherto been taught, the use thereof followeth: unto which in general is required: First, a just distance. For the sight is not infinite. Secondly, that one eye be closed: For the optic faculty conveyed from both the eyes into one, doth aim more certainly; and the instrument is more fitly applied and set to the cheek bone, then to any other place. For here the eye is as it were the centre of the circle, into which the transome is inscribed. Thirdly, the hands must be steady; for if they shake, the proportion of the Geodesy must needs be troubled and uncertain. Lastly, the place of the station is from the midst of the foot. 5 If the sight do pass from the beginning of one shank, it passeth by the end of the other: And the one shank is perpendicular unto the magnitude to be measured, the other parallel. These common and general things are premised. That the sight is from the beginning of the Index by the end of the transome: Or chose, From the beginning of the transome, unto the end of the Index. And that the Index is right, that is, perpendicular to the line to be measured, the transome parallel. Or chose. Now the perpendicularity of the Index, in measurings of lengthts, may be tried by a plummet of lead appendent● But in heights and breadths, the eye must be trusted; although a little varying of the plummet can make no sensible error. By the end of the transome, understand that which is made by the line visual, whether it be the outmost fin, or the Cursour in any other place whatsoever. 6 Length and Altitude have a threefold measure; The first and second kind of measure require but one distance, and that by granting a dimension of one of them, for the third proportional: The third two distances, and such only is the dimension of Latitude. Geodesy of right lines is two fold; of one distance, or of two. Geodesy of one distance is when the measurer for the finding of the desired dimension doth not change his place or standing. Geodesy of two distances is when the measurer by reason of some impediment lying in the way between him and the magnitude to be measured, is constrained to change his place, and make a double standing. Here observe, That length and height, may be jointly measured both with one, and with a double station: But breadth may not be measured otherwise than with two. 7 If the sight be from the beginning of the Index r●ght or plumb unto the length, and unto the father end of the same, as the segment of the Index is, unto the segment of the transome, so is the height of the measurer unto the length. The same manner of measuring shall be used form an higher place; as out of y, the segment of the Index is 5. parts; the segment of the transome 6: and then the height be 10 foot: the same Length shall be found to be 12 foot. Neither is it any matter at all, whether the length in a plain or level underneath: Or in an ascent or descent of a mountain, as in the figure under written. Thus mayest thou measure the breadths of Rivers, Valleys, and Ditches. For the Length is always after this manner, so that one may measure the distance of ships on the Sea, as also Thales Milesius, in Proclus at the 26 pj, did measure them. An example thou hast here. Hereafter in the measuring of Longitude and Altitude, fight is unto the top of the height. Which here I do now forewarn thee of, lest afterward it should in vain be reitered often. The second manner of measuring a Length is thus: 8. If the sight be from the beginning of the index parallel to the length to be measured, as the segment of the transome is, unto the segment of the index, so shall the height given be to the length. As if the segment of the Transome be 120 parts: the height given 400-foote: The segment of the Index 210 parts: The length, by the golden rule shall be 700 foot. The figure is thus. And the demonstration is like unto the former; or indeed more easier. For the triangles are equiangles, as afore. Therefore as o u is to u a: so is e i to i a. This is the first and second kind of measuring of a Longitude, by one single distance or station: The third which is by a double distance doth now follow. Here the transome, if there be room enough for the measurer to go far enough back, must be put lower, in the second distance. 9 If the sight be from the beginning of the transverie parallel to the length to be measured, as in the index the difference of the greater segment is unto the lesser; so is the difference of the second station unto the length. This kind of Geodaesy is somewhat more subtle than the former were. The figure is thus; in which let the first aiming be from a, the beginning of the transome, and out of a i the length sought by o, the end of the Index, unto e, the top of the height: And let the segment of the Index be o u: The second aiming let it be from y, the beginning of the transome, out of a greater distance by s, the end of the Index, unto e, the same note of the height: And let the segment of the Index be s r. Here the measuring performed, is the taking of the difference between o u and s r. The rest are feigned only for demonstrations sake. Therefore in the first station let a m l, be from the beginning of the transome, be parallel to the e. Here first m u, is equal to s r. For the triangle● m u a, and s r y, are equal in their shanks u a, and r y, by the grant (Because the transome standeth still in his own place:) And the angles at m u a, u a m, are equal to the angles: And all right angles are equal, by the 14 e iij. These are the outer and inner opposite one to another: And such are equal by the 1. e. v.) Therefore they are equilaters, by the 2 e seven; And o m, is the difference of the segments of the Index. Then as o m is to m u, so is e l, to l i; as the equation of three degrees doth show. For, by the 12 e seven, as o m is to m a: so is e l to l a: And as m a is to m u; so is l a, to l i. Therefore by right, as o m, is to m u: so is e l, to l i: And by the 12 e v j, so is y a, to a i: As if the difference of the first segment be 36 parts: The second segment be 72 parts: The difference of the second station 40 foot. The length sought shall be 80 foot. And here indeed is no height definitely given, that may make any bound of the principal proportion. Notwithstanding the Height, although it be of an unknown measure, is the bound of the length sought: And therefore it is an help and means to argue the question. Because it is conceived to stand plumb upon the outmost end of the length. Therefore that third kind of measuring of length is oftentimes necessary, when by neither of the former waye● the length may possibly be taken, by reason of some impediment in the way, to wit of a wall, or tree, or house, or mountain, whereby the end of the length may not be seen, which was the first way: Nor an height next adjoining to the end of the length is given, which is the second way. Hitherto we have spoken of the threefold measure of longitude, the first and second out of an height given the third out of a double distance: The measuring of height followeth next, and that is also threefold. Now height is a perpendicular line falling from the top of the magnitude, unto the ground or plain whereon the measurer doth stand, after which manner Altitude on height was defined at the 9 e iiij. The first geodesy or manner of measuring of heights is thus. 10. If the sight be from the beginning of the transome perpendicular unto the height to be measured, as the segment of the transome, is unto the segment of the Index, so shall the length given be to the height. Let the segment of the transome be 60 parts: the segment of the Index 36: the Length given 120 foot: the height sought shall be, by the golden rule, 72 foot. The Figure is thus: And the demonstration is by the 12 e seven, as afore: but here is to be added the height of the measurer; which if it be 4 foot, the whole height shall be 76 foot. Therefore in an eversed altitude 11. If the sight be from the beginning of the Index parallel to the height, as the segment of the transome is, unto the segment of the index, so shall the length given be, unto the height sought. Eversa altitudo, An eversed altitude (Reversed, H;) is that which we call depth, which indeed is nothing else, in the Geometers sense, but height turned topsy turvy, as we say, or with the heels upward. For out of the height concluded by subducting that which is above ground, the height or depth of a Well shall remain. 12. If the sight be from the beginning of the Index perpendicular to the height to be measured, as the segment of the Index is unto the segment of the Transome, so shall the length given be to the height. Therefore 13. If the sight be from the beginning of the Index (perpendicular to the magnitude to be measured) by the names of the transome, unto the ends of some known part of the height, as the distance of the Names is, unto the rest of the transome above them, so shall the known part be unto the part sought. Or thus: If the sight pass from the beginning of the Index being right, by the vanes of the transversary, to the terms of some parts; as the distance of the vanes is unto the rest of the transversary above the index, so is the part known unto the remainder: H. This is a consectary of a known part of an height, from whence the rest may be known, as in the figure. The first and second kind of measuring of heights is thus: The third followeth. 14 If the sight be from the beginning of the Index perpendicular to the height, as in the Index the difference of the segmeut, is unto the difference of the distance or station; so is the segment of the transome unto the height. Hitherto you must recall that subtlety, which was used in the third manner of measuring of lengths. Let the first aim be taken from a, the beginning of the Index perpendicular unto the height to be measured: And from an unknown length a i, by o, the end of the transome, unto e, the top of the height e i: And let the segment of the Index be u a. The second aim, let it be taken from y, the beginning of the same Index; and out of a greater distance, by s, the end of the transome, unto the same top e. And the segment of the Index let it be r l. Here, as afore, the measuring is performed and done, by the taking of the difference of the said y r, above a u: Now the demonstration is concluded, as in the former was taught. Let the parallel l s m, be erected against a o e. Here first the triangles o u a, & s r l, are equilaters, by the 2 e seven.; (seeing that the angles at a, and l, the external and internal, are equal in bases o u, and s r: for the segment in each distance is the same still:) Therefore u a, is equal to r l. Now the rest is concluded by a sorites of four degrees: As y r, is unto y u: so by the 12. e seven. is s r, that is, o u, unto e i: And as o u, is unto e i, so is a u, that is, l r, unto a i. Therefore the remainder y l, unto the remainder y a; shall be as y r, is unto the whole y i, and therefore from the first unto the last, as s r, is to e i. Therefore let the difference of the Index be 23: partest The difference of the distance 30. foot: The segment of the transome 23. parts: The height shall be 57 9/23. or foot. Therefore 15 Out of the Geodesy of heights, the difference of two heights is manifest. Or thus: By the measure of one altitude, we may know the difference of two altitudes: H. For when thou hast taken or found both of them, by some one of the former ways, take the lesser out of the greater; and the remain shall be the height desired. From hence therefore by one of the towers of unequal height, you may measure the height of the other. First out of the lesser, let the length be taken by the first way: Because the height of the lesser, wherein thou art, is easy to be taken, either by a plumbe-line, let fall from the top to the bottom, or by some one of the former ways. Then measure the height, which is above the lesser: And add that to the lesser, and thou shalt have the whole height, by the first or second way. The figure is thus, and the demonstration is out of the 12. e seven. For as a e, is to e i, so is a o, to o u. chose out of an higher Tower, one may measure a lesser. 16 If the sight be first from the top, than again from the base or middle place of the greater, by the vanes of the transome unto the top of the lesser height; as the said parts of the yards are unto the part of the first yard; so the height between the stations shall be unto his excess above the height desired. For let the parts of the yard be 12. and 6. and the sum of them 18. Now as 18. is 12. so is the whole altitude u y, 190. foot, unto the excess 126⅔ foot. The remainder therefore 63⅓ foot, shall be a s, the lesser height sought. The second station might have been in o, the end of the perpendicular from a. But by taking the aim out of the top of the lesser altitude, the demonstration shall be yet again more easy and short, by the two triangles at the yard a e i, and a e f, resembling the two whole triangles a o u, and a o y, in like situation, the parts of the shank cut, are on each side the segments of the transome. One may again also out of the top of a Turret measure the distance of two turrets one from another: For it is the first manner of measuring of longitudes, neither doth it here differ any whit from it, more than the yard is hanged without the height given. The figure is thus: And the Demonstration is by the 12. e seven. For as a e, the segment of the yard, is unto e i, the segment of the transome: so is the assigned altitude a o, unto the length oh u. The geodesy or measuring of altitude is thus, where either the length, or some part of the length is given, as in the first and second way: Or where the distance is double, as in the third. 17 If the sight be from the beginning of the yard being right or perpendicular, by the vanes of the transome, unto the ends of the breadth; as in the yard the difference of the segment is unto the difference of the distance, so is the distance of the vanes unto the breadth. The measuring of breadth, that is, of a thwart or cross line, remaineth. The Figure and Demonstration is thus: The first aiming, let it be a e i, by o, and u, the vanes of the transome o u. The second, let it be y e i, by s, and r, the vanes of the transome s r. Then by the point s, let the parallel l s m, be drawn against a o e. Here first, the triangles o u a, and s i l, are equilaters, by the 2 e seven. Because the angles at n and j, are right angles: And u a o, and j l s, the outer and inner, are equal in their bases o u, and s j, by the grant: Because here the segment of the transome remaineth the same: Therefore u a, is equal to j l. These grounds thus laid, the demonstration of the third altitude here taken place. For as y l, is unto y a: so is s j, unto e r: And, because parts are proportional unto their multiplicants, so is s r, unto e i: for the rest do agree. The same shall be the geodesy or manner of measuring, if thou wouldst from some higher place, measure the breadth that is beneath thee, as in the last example. But from the distance of two places, that is, from latitude or breadth, as of Trees, Mountains, Cities, Geographers and Chorographers do gain great advantages and helps. The tenth Book of Geometry, of a Triangulate and Parallelogramme. ANd thus much of the geodesy of right lines, by the means of rectangled triangles: It followeth now of the triangulate. 1. A triangulate is a rectilineall figure compounded of triangles. As before (for the dichotomies sake) of a line was made a Lineate, to signify the genus of a surface and a Body: so now is for the same cause of a triangle made a Triangulate, to declare and express the genus of a Quadrilater and Multilater, and indeed more justly, then before in a Lineate. For triangles do compound and make the triangulate, but lines do not make the lineate. Therefore 2. The sides of a triangulate are two more than are the triangles of which it is made. And 3. Homogeneal Triangulates are cut into an equal number of triangles, è 20 p vi. For if they be Quadrangles, they be cut into two triangles: If Quinquangles, into 3. If Hexangles, into 4, and so forth. 4. Like triangulates are cut into triangles alike one to another and homologal to the whole è 20 p vi. Or thus: Like Triangulates are divided into triangles like one unto another, and in porportion correspondent unto the whole: H. As in these two quinqualges. First the particular triangles are like between themselves. For the shanks of a e u and y s m, equal angles are proportional, by the grant. Therefore the triangles themselves are equiangles, by 14 e seven. And therefore alike, by the 12 e seven. and so forth of the rest. The middle triangles, the equal angles being substracted shall have their other angles equal: And therefore they also shall be equiangles and alike, by the same. Secondarily, the triangles a e u, and y s m: e i o and s r l; e o u, and s l m, to wit, alike between themselves, are by the 1 e uj, in a double reason of their homologal sides e u, s m, e o, s l, which reason is the same, by means of the common sides. Therefore three triangles are in the same reason: And therefore they are proportional: And, by the third composition, as one of the antecedents is, unto one of the consequents● so is the whole quinquangle to the whole. 5. A triangulate is a Quadrangle or a Multangle. The parts of this partition are in Euclid, and yet without any show of a division. And here also, as before, the species or several kinds have their denomination their angles, although it had been better and truer to have been taken from their sides; as to have been called a Quadrilater, or a Multilater. But in words use must be followed as a master. 6. A Quadrangle is that which is comprehended of four right lines. 22 d i. 7. A quadrangle is a a Parallelogramme, or a Trapezium. This division also in his parts is in the Elements of Euclid, but without any form or show of a division. But the difference of the parts shall more fitly be distinguished thus: Because in general there are many common parallels. 8. A Parallelogramme is a quadrangle whose opposite sides are parallel. Therefore 9 If right lines on one and the same side, do jointly bound equal and parallall lines, they shall make a parallelogramme. The reason is, because they shall be equal and parallel between themselves, by the 26. e v. And 10 A parallelogramme is equal both in his opposite sides, and angles, and segments cut by the diameter. Or thus; The opposite, both sides, and angles, and segments cut by the diameter are equal. Three things are here concluded; The first is, that the opposite sides are equal: This manifest by the 26 e v. Because two right lines do jointly bound equal parallels. And 11 The Diameter of a parallelogramme is cut into two by equal rays. As in the three figures a e i, next before: This a parallelogramme hath common with a circle, as was manifest at the 28. e iiij. And 12 A parallelogramme is the double of a triangle of a trinangle of equal base and height, 41. p i. And 13 A parallelogramme is equal to a triangle of equal height and double base unto it: è 42. p i. From whence one may 14 To a triangle given, in a rectilineall angle given, make an equal parallelogramme. 15 A parallelogramme doth consist both of two diago●als, and compliments, and gnomon. For these three parts of a parallelogramme are much used in Geometrical works and businesses, and therefore they are to be defined. 16 The diagonal is a particular parallelogramme having both an angle and diagonal diameter common with the whole parallelogramme. 17 The diagonal is like, and alike situate to the whole parallelogramme: è 24. p vi. There is not any, either rate or proportion of the diagonal propounded, only similitude is attributed to it, as in the same figure, the diagonal a u y s, is like unto the whole parallelogramme a e i o. For first it is equianglar to it. For the angle at a, is common to them both: And that is equal to that which is at y, (by the 10. e x:) And therefore also it is equal to that at i, by the 10. e x. Then the angel's a u y, and a s y, are equal, by the 21. e v. to the opposite inner angles at e, and o. Therefore it is equiangular unto it. Again, it is proportional to it in the shanks of the equal angles. For the triangles a u y, and a e i, are alike, by the 12 e seven, because u y is parallel to the base. Therefore as a u is to u y; so is a i to e i: Then as u y is to y a; so is e i to i a. Again by the 21 e v, because s y is parallel to the base i o, as a y is to y s: so is a i, to i o: Therefore equiordinately, as u y is to y s: so is e i to i o: Item as s y is to y a, so is i o to i a: And as y a is to a s: so is i a to a o. Therefore equiordinately, as y s is to s a: so is i o to o a. Lastly as s a is unto a y; so is o a unto a i: And as a y is to a u; so is a i unto a e. Therefore equiordinately, as s a is to a u: so is a o, to a e. Wherefore the diagonal s u is proportional in the shanks of equal angles to the parallelogramme oh e. The demonstration shall be the same of the diagonal r l. The like situation is manifest, by the 21 e iiij. And from hence also is manifest, That the diagonal of a Quadrate, is a Quadrate: Of an Oblong, an Oblong: Of a rhomb, a rhomb: Of a Rhomboides, a Rhomboides: because it is like unto the whole, and a like situate. Now the diagonals seeing they are like unto the whole and a like situate, they shall also be like between themselves and alike situate one to another, by the 21 and 22 e iiij. Therefore 18. If the particular parallelogramme have one and the same angle with the whole, be like and alike situate unto it, it is the diagonal. 26 p vi. As for example, Let the particular parallelogramme a u y s, be coangular to the whole parallelogramme a e i o: And let it have the same angle with it at a; like unto the whole and alike situate unto it; I say it is the diagonal. Otherwise, let the divers Diagony be a r o: And let l r be parallel against a e: Therefore a l r s, shall be the diagonal, by the 6 e [15.] Now therefore it shall be, by 8 e [16 e,] as e a is to a i: so is s a unto a l: Again, by the grant, as e a is unto a i: so is s a to a u: Therefore the same s a is proportional to a l, and to a u: And a l is equal to a u, the part to the whole, which is impossible. 19 The Compliment is a particular parallelogramme, comprehended of the conterminall sides of the diagonals. Or thus: It is a particular parallelogramme contained under the next adjoining sides of the diagonals. 20. The compliments are equal. 43 p i. Therefore 21. If one of the Compliments be made equal to a triangle given, in a rightlined angle given, the other made upon a right line given shall be in like manner equal to the same triangle. 44 p i. As if thou shouldest desire to have a parallelogramme upon a right line given, and in a right lined angle given, to be made equal to a triangle given, this proposition shall give satisfaction. And 22 If parallelogrammes be continually made equal to all the triangles of an assigned triangulate, in a right lined angle given, the whole parallelogramme shall in like manner be equal to the whole triangulate. 45 p i. This is a corollary of the former, of the Reason or rate of a Parallelogramme with a Triangulate; and it needeth no father demonstration; but a ready and steady hand in describing and working of it. Here thou hast 3 compliments continued, and continuing the Parallelogramme: But it is best in making and working of them, to put out the former, and one of the sides of the inferior or latter diagonal, lea●t the confusion of lines do hinder or trouble thee. Therefore 23. A Parallelogramme is equal to his diagonals and compliments. For a Parallelogramme doth consist of two diagonals, and as many compliments: Wherefore a Parallelogramme is equal to his parts: And again the parts are equal to their whole. 24. The Gnomon is any one of the diagonal with the two compliments. In the Elements of Geometry there is no other use, as it seemeth of the gnomon than that in one word three parts of a parallelogramme might be signified and called by three letters a e i. Otherwise gnomon is a perpendicular. 25. parallelograms of equal height are one to another as their bases are. 1 p vi. Therefore 26 Parallelogrammes of equal height upon equal bases are equal. 35. 36 p i. As is manifest in the same example. 27 If equiangle parallelogrammes be reciprocal in the shanks of the equal angle, they are equal: And chose. 15 p vi. Therefore 28 If four right lines be proportional, the parallelogramme made of the two middle ones, is equal to the equiangled parallelogramme made of the first and last: And chose, e 16 p vi. For they shall be equiangled parallelogrammes reciprocal in the shanks of the equal angle. And 29 If three right lines be proportional, the parallelogramme of the middle one is equal to the equiangled parallelogramme of the extremes: And chose. It is a consectary drawn out of the former. Of Geometry, the eleventh Book, of a Right angle. 1. A Parallelogramme is a Right angle or an Obliquangle. HItherto we have spoken of certain common and general matters belonging unto parallelogrammes● specials do follow in Rectangles and Obliquangles, which difference, as is aforesaid, is common to triangles and triangulates. But at this time we find no fitter words whereby to distinguish the generals. 2. A Right angle is a parallelogramme that hath all his angles right angles. As in a e i o. And here hence you must understand by one right angle that all are right angles. For the right angle at a, is equal to the opposite angle at i, by the 10 e x. Therefore 3 A rightangle is comprehended of two right lines comprehending the right angle 1. d ij. Comprehension, in this place doth signify a certain kind of Geometrical multiplication. For as of two numbers multiplied between themselves there is made a number: so of two sides (ductis) driven together, a right angle is made: And yet every right angle is not rational, as before was manifest, at the 12. e iiij. and shall after appear at the 8 e. And 4 Four right angles do fill a place. Neither is it any matter at all whether the four rectangles be equal, or unequal; equilaters, or unequilaters; homogeneals, or heterogenealls. For which way so ever they be turned, the angles shall be right angles: And therefore they shall fill a place. 5 If the diameter do cut the side of a right angle into two aquall parts, it doth cut it perpendicularly: And chose. Therefore 6 If an inscribed right line do perpendicularly cut the side of the right angle into two equal parts, it is the diameter. The reason is, because it doth cut the parallelogramme into two equal portions. 7 A right angle is equal to the rightangles made of one of his sides and the segments of the other. As here the four particular right angles are equal to the whole, which are made of a e, one of his sides, and of e i, i o, o u, u y, the segments of the other. Lastly, every arithmetical multiplication of the whole numbers doth make the same product, that the multiplication of the one of the whole numbers given, by the parts of the other shall make; yea, that the multiplication of the parts by the parts shall make. This proposition is cited by Ptolomey in the 9 Chapter of the 1 book of his Almagest. 8 If four right lines be proportional, the rectangle of the two middle ones, is equal to the rectangle of the two extremes. 16. p vi. 9 The figurate of a rational rectangle is called a rectinall plain. 16. d seven. If therefore the Base of a Rectangle be 6. And the height 4. The plot or content shall be 24. And if it be certain that the rectangles content be 24. And the base be 6. It shall also be certain that the height is 4. The example is thus. This manner of multiplication, say 1, is Geometrical: Neither are there here, of lines made lines, as there of unities were made unities; but a magnitude one degree higher, to wit, a surface, is here made. Here hence is the Geodesy or manner of measuring of a rectangled triangle made known unto us. For when thou shalt multiply the shanks of a right angle, the one by the other, thou dost make the whole rectangled parallelogramme, whose half is a triangle, by the 12. e x. Of Geometry the twelfth Book, Of a Quadrate. 1 A Rectangle is a Quadrate or an Oblong. THis division is made in proper terms: but the thing itself and the subject difference is common out of the angles and sides. 2 A Quadrate is a rectangle equilater 30. dj. Plains are with us, according to their divers natures and qualities, measured with divers and sundry kinds of measures. Board, Glass, and Paving-stone are measured by the foot: Cloth, Wainscot, Painting, Paving, and such like, by the yard: Land, and Wood, by the Perch or Rodde, Of Measures● and the sundry sorts thereof commonly used and mentioned in histories we have in the former spoken at large: Yet for the farther confirmation of some thing then spoken, and here again now upon this particular occasion repeated, it shall not be amiss to hear what our Statutes speak of these three sorts here mentioned. It is ordained, saith the Statute, That three Barley-cornes dry and round, do make an Inch: twelve ynches do make a Foot: three foot do make a Yard: Five yards and an half do make a Perch: Forty perches in length, and four in breadth do make an Acre. 33. Edwardi 1. De Terris mensurandis. Item, De compositione Vlnarum & Perticarum. Moreover observe, that all those measures there spoken of were only lengths: These here now last repeated, are such as the magnitudes by the measured are, in Planimetry, I mean, they are Plains: In Stereometry they are solids, as hereafter we shall make manifest. Therefore in that which followeth, An inch is not only a length three barley-cornes long: but a plain three barley-cornes long, and three broad. A Foot is not only a length of 12. ynches: But a plain also of 12. ynches square, or containing 144. square ynches● A yard is not only the length of three foot: But it is also a plain 3. foot square every way. A Perch is not only a length of 5½. yards: But it is a plot of ground 5½. yards square every way. A Quadrate therefore or square, seeing that it is equilater that is of equal sides: And equiangle by means of the equal right angles, of quandrangles that only is ordinate. Therefore 3 The sides of equal quadrates, are equal. And The sides of equal quadrates are equally compared: If therefore two or more quadrates be equal, it must needs follow that their sides are equal one to another. And 4 The power of a right line is a quadrate. Or thus: The possibility of a right line is a square H. A right line is said posse quadratum, to be in power a square; because being multiplied in itself, it doth make a square. 5 If two conterminall perpendicular equal right lines be closed with parallels, they shall make a quadrate. 46. p.j. 6 The plain of a quadrate is an equilater plain. Or thus: The plain number of a square, is a plain number of equal sides, H. A quadrate or square number, is that which is equally equal: Or that which is comprehended of two equal numbers, A quadrate of all plains is especially rational; and yet not always: But that only is rational whose number is a quadrate. Therefore the quadrates of numbers not quadrates, are not rationals. Therefore 7 A quadrate is made of a number multiplied by itself. Such quadrates are the first nine. 1,4,9, 16,25,36,49,64, 81, made of once one, twice two, thrice three, four times four, five times five, six times six, seven times seven, eight times eight, and nine times nine. And this is the sum of the making and invention of a quadrate number of multiplication of the side given by itself. Hereafter divers comparisons of a quadrate or square, with a rectangle, with a quadrate, and with a rectangle and a quadrate jointly. The comparison or rate of a quadrate with a rectangle is first. 8 If three right lines be proportional, the quadrate of the middle one, shall be equal to the rectangle of the extremes: And chose: 17. p v i. and 20. p seven. It is a corollary out of the 28. e x. As in a e, e i, i o. 9 If the base of a triangle do subtend a right angle, the power of it is as much as of both the shanks: And chose 47,48. p i. 10 If the quadrate of an odd number, given for the first shank, be made less by an unity; the half of the remainder shall be the other shank; increased by an unity it shall be the base. Or thus: If the square of an odd number given for the first foot, have an unity taken from it, the half of the remainder shall be the other foot, and the same half increased by an unity, shall be the base: H. Again, the quadrate or square of 3. the first shank is 9 and 9— 1. is 8, whose half 4, is the other shank. And 9— 1, is 10. whose half 5. is the base. Plato's way is thus by an even number. 11 If the half of an even number given for the first shank be squared, the square number diminished by an unity shall be the other shank, and increased by an unity it shall be the base. Again, the quadrate or square of 3. the half of 6, the first shank, is 9 and 9— 1, is 8, for the second shank. And out of this rate of rational powers (as Vitruvius, in the 2. Chapter of his IX. book) saith Pythagoras taught how to make a most exact and true squire, by joining of three rulers together in the form of a triangle, which are one unto another as 3,4. and 5. are one to another. From hence Architecture learned an Arithmetical proportion in the parts of ladders and stairs. For that rate or proportion, as in many businesses and measures is very commodious; so also in buildings, and making of ladders or stairs, that they may have moderate rises of the steps, it is very speedy. For 9— 1. is, 10, base. 12. The power of the diagony is twice as much, as is the power of the side, and it is unto it also incommensurable. Or thus: The diagonal line is in power double to the side, and is incommensurable unto it, H. This is the way of doubling of a square taught by Plato, as Vitruvius telleth us: Which notwithstanding may be also doubled, trebled, or according to any reason assigned increased, by the 25 e iiij, as there was foretold. But that the Diagony is incommensurable unto the side it is the 116 p x. The reason is, because otherwise there might be given one quadrate number, double to another quadrate number: Which as Theon and Campanus teach us, is impossible to be found. But that reason which Aristotle bringeth is more clear which is this; Because otherwise an even number should be odd. For if the Diagony be 4, and the side 3: The square of the Diagony 16, shall be double to the square of the side: And so the square of the side shall be 8. and the same square shall be 9, to wit, the square of 3. And so even shall be odd, which is most absurd. Hither may be added that at the 42 p x. That the segments of a right line diversely cut; the more unequal they are the greater is their power. 13 If the base of a right angled triangle be cut by a perpendicular from the right angle in a doubled reason, the power of it shall be half as much more, as is the power of the greater shank: But thrice so much as is the power of the lesser. If in a quadrupled reason, it shall be four times and one fourth so much as is the greater: But five times so much as is the lesser, At the 13, 15, 16 p x iij. And by the same argument it shall be treble unto the quadrate or square of e i. The other, of the fourfold or quadruple section, are manifest in the figure following, by the like argument. 14 If a right line be cut into how many parts so ever, the power of it is manifold unto the power of segment, denominated of the square of the number of the section. Or thus: if a right be cut into how many parts so ever it is in power the multiplex of the segment, the square of the number of the section, being denominated thereof: H. 15. If a right line be cut into two segments, the quadrate of the whole is equal to the quadrats of the segments, and a double rectanguled figure, made of them both. 4 p ij. The third rate of a quadrate is hereafter with two rectangles, and two quadrates, and first of equality. Therefore 16. The side of the first diagonal, is the side of one of the compliments; And being doubled, it is the side of them both together: Now the other side of the same compliments both together, is the side of the other diagonal. The side of a quadrate given is many times in numbers sought. Therefore by the former element and his consectaries the resolution of a quadrates side is framed and performed. This is the rate of a quadrate with a rectangle & a quadrate from whence is had the analisis or resolution of the side of a quadrate expressable by a number. For it is the same way from Cambridge to London, that is from London to Cambridge. And this use of geometrical analysis remaineth, as afterward in a Cube, when as otherwise through the whole book of Euclides Elements there is no other use at all of that. Then beginning at the left hand, as in Division, Or thus: that is where we left in multiplication, and I seek amongst the squares the greatest contained in the first period, which here is 1● And the side of it, which is also 1, I place with my quotient: Then I square this quotient, that is I multiply it by itself, and the product 1, I sect under the same first period: Lastly, I subtract it from the same period, and there remaineth not any thing. Then as in division I set up the figures of the next period one degree higher. Secondly double the side now found, and it shall be 2, which I place in manner of a Divisor, on the left hand, within the semicircle: By this I divide the 40, the two compliments or Plains, and I find the quotient or second side 2; which I place in the quotient by 1, This side I multiply first quadrate like, that is by itself; and I make 4, the lesser diagonal: And therefore I place under the last 4: Then I multiply the said Divisor 2, by the same 2 the quotient, Or thus: and I make in like manner, 4 which I place under the dividend, or the first 4. Lastly I subtract these products from the numbers above them, and remaineth nothing. Therefore I say first, That 144, the number given is a quadrate: And moreover, That 12 is the true side of it. Again, let the side of 15129 be sought. First divide it into imperfect periods as before was taught; in this manner: 15129. Then I seek amongst the former quadrates, for the side of 1, the quadrate of the first period; and I find it to be 1: This side I place within the quotient or lunular on the right side: Lastly I subtract 1 from ●, and nothing remaineth. Then I double the said side found; and I make 2: This 2, I place for my divisor within the lunular or semicircle on the left hand: By which I divide 5; And I find the quotient 2, which I place by the former quotient: Then I multiply the same 2, first quadratelike by itself, and I make 4. Then I multiply the said divisour by 2, the quotient, and I make likewise 4: which I place underneath 51. Lastly, I subtract the same 44, from 51, and there remain 7, over the head of 1; By which I place 29, the last period remaining. Again I double 12, my whole quotient, and I make 24. By this double I divide 72, the double Compliment remaining, and I find 3 for the side or quotient: First this side I multiply quadratelike by itself, and I make 9, which I place underneath 9, the last figure of my dividende. Then again, by the same quotient, or side 3, I multiply 2●: my divisour, and I make 72; which I place under 72, the two figures of my Dividende: Lastly I subtract the under figures, from the upper, and there is likewise nothing remaining: Wherefore I say, as afore; that the figurate 15129 given, is a square: And the side thereof is 123. Sometime after the quadrate now found, in the next places, there is neither any plain nor square to be found: Therefore the single side thereof shall be O. As in the quadrate 366025, the whole side is 605, consisting of three several sides, of which the middle one is o. Sometime also the middle plain doth contain a part of the quadrate next following: Therefore if the other side remaining be greater than the side of the quadrate following, it is to be made equal unto it: As for example, Let the side of the quadrate 784, be sought; Or thus: The side of the first quadrate shall be 2, and there shall remain 3, thus: Then the same side doubled is 4 for the quotient; Which is found in 38, the double plain remaining 9 times, for the other side: But this side is greater than the side of the next following quadrate: Take therefore 1 out of it: And for nine take 8, and place it in your quotient; Which 8 multiplied by itself maketh 64, for the Lesser quadrate: And again the same multiplied by 4 the divisour maketh 32; The sum of which two products 384, subtracted from the remain 384, leave nothing: Therefore 784 is a Quadrate: And the side is 28. And from hence the invention of a mean proportionll, between two numbers given, (if there be any such to be found) is manifest. For it the product of two numbers given be a quadrate, the side of the quadrate shall be the mean proportional, between the numbers given; as is apparent by the golden rule: As for example, Between 4. and 9 two numbers given, I desire to know what is the mean proportion. I multiply therefore 4 and 9 between themselves, and the product is 36: which is a quadrate number; as you see in the former; And the side is 6. Therefore I say, the mean proportional between 4. and 9 is 6, that is, As 4. is to 6. so is 6. to 9 If the number given be not a quadrate, there shall no arithmetical side, and to be expressed by a number be found: And this figurate number is but the shadow of a Geometrical figure, and doth not indeed express it fully, neither is such a quadrate rational: Yet notwithstanding the numeral side of the greatest square in such like number may be found: As in 148. The greatest quadrate continued is 144 and the side is 12. And there do remain 4. Therefore of such kind of number, which is not a quadrate, there is no true or exact side: Neither shall there ever be found any so near unto the true one; but there may still be one found more near the truth. Therefore the side is not to be expressed by a number. Of the invention of this there are two ways: The one is by the Addition of the gnomon; The other is by the Reduction of the number assigned unto parts of some greater denomination. The first is thus: 17 If the side found be doubled, and to the double a unity be added, the whole shall be the gnomon of the next greater quadrate. For the sides is one of the compliments, and being doubled it is the side of both together. And an unity is the latter diagonal. So the side of 148 is 12 4/25. The reason of this dependeth on the same proposition, from whence also the whole side, is found. For seeing that the side of every quadrate lesser than the next follower differeth only from the side of the quadrate next above greater than it but by an 1. the same unity, both twice multiplied by the side of the former quadrate, and also once by itself, doth make the Gnomon of the greater to be added to the quadrate. For it doth make the quadrate 169. Whereby is understood, that look how much the numerator 4. is short of the denominatour 25. so much is the quadrate 148. short of the next greater quadrate. For it thou do add 21. which is the difference whereby 4 is short of 25. thou shalt make the quadrate 169. whose side is 13. The second is by the reduction, as I said, of the number given unto parts assigned of some great denomination, as 100 or 1000 or some smaller than those, and those quadrates, that their true and certain may be known: Now look how much the smaller they are, so much nearer to the truth shall the side found be. Moreover in lesser parts, the second way beside the other, doth show the side to be somewhat greater than the side, by the first way found: as in 7. the side by the first way is 3/25. But by the second way the side of 7. reduced unto thousands quadrates, that is unto 7000000/1000000, that is, 2645/1000, and beside there do remain 3975. But 645/1000. are greater than 3/5. For ⅗. reduced unto 1000 are but 600/1000. Therefore the second way, in this example, doth exceed the first by 45/1000. those remains 3975. being also neglected. Therefore this is the Analysis or manner of finding the side of a quadrate, by the first rate of a quadrate, equal to a double rectangle and quadrate. The Geodesy or measuring of a Triangle. There is one general Geodesy or way of measuring any manner of triangle whatsoever in Hero, by addition of the sides, halving of the sum, subduction, multiplication, and invention of the quadrates side, after this manner. 18 If from the half of the sum of the sides, the sides be severally subducted, the side of the quadrate continually made of the half, and the remains shall be the content of the triangle. This general way of measuring a triangle is most easy and speedy, where the sides are expressed by whole numbers. The special geodesy of rectangle triangle was before taught (at the 9 e x i.) But of an oblique angle it shall hereafter be spoken. But the general way is far more excellent than the specially For by the reduction of an obliquangle many frauds and errors do fall out, which caused the learned Cardine merrily to wish, that he had but as much land as was lost by that false kind of measuring. 19 If the base of a triangle do subtend an obtuse angle, the power of it is more than the power of the shanks, by a double right angle of the one, and of the continuation from the said obtusangle unto the perpendicular of the top. 12. p ij. Or thus: If the base of a triangle do subtend an obtuse angle, it is in power more than the feet, by the right angled figure twice taken, which is contained under one of the feet and the line continued from the said foot unto the perpendicular drawn from the top of the triangle. H. There is a comparison of a quadrate with two in like manner triangles, and as many quadrates, but of unequality. For by 9 e, the quadrate of a i, is equal to the quadrates of a o, and o ay, that is, to three quadrates of i o, oh e, e a, and the double rectangle aforesaid. But the quadrates of the shanks a e, e i, are equal to those three quadrates, to wit, of a i, his own quadrate, and of e i, two, the first i o, the second o e, by the 9 e. Therefore the excess remaineth of a double rectangle. Of Geometry, the thirteenth Book, Of an Oblong. 1 An Oblong is a rectangle of inequal sides, 31. d i. This second kind of rectangle is of Euclid in his elements properly named for a definitions sake only. The rate of Oblongs is very copious, out of a threefold section of a right line given, sometime rational and expresable by a number: The first section is as you please, that is, into two segments, equal or unequal: From whence a fivefold rate ariseth. 2 An Oblong made of an whole line given, and of one segment of the same, is equal to a rectangle made of both the segments, and the square of the said segment. 3. p ij. It is a consectary out of the 7 e xj. For the rectangle of the segments, and the quadrate, are made of one side, and of the segments of the other. Now a rectangle is here therefore proposed, because it may be also a quadrate, to wit, if the line be cut into two equal parts. Secondarily, 3 Oblongs made of the whole line given, and of the segments, are equal to the quadrate of the whole 2 p ij. This is also a Consectary out of the 4. e xj. Here the segments are more than two, and yet notwithstanding from the first the rest may be taken for one, seeing that the particular rectangle in like manner is equal to them. This proposition is used in the demonstration of the 9 e xviij. Thirdly, 4 Two Oblongs made of the whole line given, and of the one segment, with the third quadrate of the other segment, are equal to the quadrates of the whole, and of the said segment. 7 p ij. 5 The base of an acute triangle is of less power than the shanks are, by a double oblong made of one of the shanks, and the one segment of the same, from the said angle, unto the perpendicular of the top. 13. p.ij. And from hence is had the segment of the shank toward the angle, and by that the perpendicular in a triangle. Therefore 6. If the square of the base of an acute angle be taken out of the squares of the shanks, the quotient of the half of the remain, divided by the shank, shall be the segment of the dividing shank from the said angle unto the perpendicular of the top. Now again from 169, the quadrate of the base 13, take 25, the quadrate of 5, the said segment: And the remain shall be 144, for the quadrate of the perpendicular a o, by the 9 e x ij. Here the perpendicular now found, and the sides cut, are the sides of the rectangle, whose half shall be the content of the Triangle: As here the Rectangle of 21 and 12 is 252; whose half 126, is the content of the triangle. The second section followeth from whence ariseth the fourth rate or comparison. 7. If a right line be cut into two equal parts, and otherwise; the oblong of the unequal segments, with the quadrate of the segment between them, is equal to the quadrate of the bisegment. 5 p ij. The third section doth follow, from whence the fifth reason ariseth. 8. If a right line be cut into equal parts; and continued; the oblong made of the continued and the continuation, with the quadrate of the bisegment or half, is equal to the quadrate of the line compounded of the bisegment and continuation. 6 p ij. From hence ariseth the Mesographus or Mesolabus of Heron the mechanic; so named of the invention of two lines continually proportional between two lines given. Whereupon arose the Deliacke problem, which troubled Apollo himself. Now the Mesographus of Hero is an infinite right line, which is stayed with a scrue-pinne, which is to be moved up and down in riglet. And it is as Pappus saith, in the beginning of his 111 book, for architects most fit, and more ready than the Plato's mesographus. The mechanical handling of this mesographus, is described by Eutocius at the 1 theorem of the 11 book of the sphere; But it is somewhat more plainly and easily thus laid down by us. 9 If the Mesographus, touching the angle opposite to the angle made of the two lines given, do cut the said two lines given, comprehending a right angled parallelogramme, and infinitely continued, equally distant from the centre, the intersegments shall be the means continually proportionally, between and two lines given. Or thus: If a Mesographus, touching the angle opposite to the angle made of the lines given, do cut the equal distance from the centre, the two right lines given, containing a right angled parallelogramme, and continued out infinitely, the segments shall be mean in continual proportion with the line given: H. As let the two right-lines given be a e, and a i: And let them comprehend the rectangled parallelogramme a o: And let the said right lines given be continued infinitely, a e toward u; and a i toward y. Now let the Mesographus u y, touch o, the angle opposite to a: And let it cut the said continued lines equally distant from the Centre. (The centre is found by the 8 e iiij, to wit, by the meeting of the diagonies: For the equidistance from the centre the Mesographus is to be moved up or down, until by the Compasses, it be found.) Now suppose the points of equidistancy thus found to be u, and y. I say, That the portions of the continued lines thus are the mean proportionals sought: And as a e is to i y: so is i y, to e u, so is e u, to a i The fourteenth Book, of P. Ramus Geometry: Of a right line proportionally cut: And of other Quadrangles, and Multangels. THus far of the threefold section, from whence we have the five rational rates of equality: There followeth of the third section another section, into two segments proportional to the whole. The section itself is first to be defined. 1. A right line is cut according to a mean and extreme rate, when as the whole shall be to the greater segment; so the greater shall be unto the lesser. 3. d vi. This line is cut so, that the whole line itself, with the two segments, doth make the three bounds of the proportioned And the whole itself is first bound: The greater segment is the middle bound: The lesser the third bound. 2. If a right line cut proportionally be rational unto the measure given, the segments are unto the same, and between themselves irrational è 6 p xiij. A Triangle, and all Triangulates, that is figures made of triangles, except a Rightangled-parallelogramme, are in Geometry held to be irrationals. This is therefore the definition of a proportional section: The section itself followeth, which is by the rate of an oblong with a quadrate. 3. If a quadrate be made of a right line given, the difference of the right line from the midst of the conterminall side of the said quadrate made, above the same half, shall be the greater segment of the line given proportionally cut: 11 p ij. Or thus: If a square be made of a right line given, the difference of a right line drawn from the angle of the square made unto the midst of the next side, above the half of the side, shall be the greater segment of the line given, being proportionally cut: H. For of y a, let the quadrate a y s r, be made: And let s r, be continued unto l. Now by the 8 e, xiij. the oblong of o y, and a y, with the quadrate of u a, is equal to the quadrate of u y, that is by the construction of u e: And therefore, by the 9 e xij. it is equal to the quadrates e a, and a u: Take away from each side the common oblong a l, and the quadrate y r, shall be equal to the oblong r i. Therefore the three right lines, e a, a r, and r e, by the 8 e xij. are continual proportional. And the right line a e, is cut proportionally. Therefore 4 If a right line cut proportionally, be continued with the greater segment, the whole shall be cut proportionally, and the greater segment shall be the line given. 5 p xiij. As in the same example, the right line oh y, is continued with the greater segment, and the oblong of the whole and the lesser segment is equal to the quadrate of the greater. And thus one may by infinitely proportionally cutting increase a right line; and again decrease it. The lesser segment of a right line proportionally cut, is the greater segment, of the greater proportionally cut. And from hence a decreasing may be made infinitely. 5 The greater segment continued to the half of the whole, is of power quintuple unto the said half, that is, five times so great as it is: and if the power of a right line be quintuple to his segment, the remainder made the double of the former is cut proportionally, and the greater segment, is the same remainder. 1. and 2. p x iij. This is the fabric or manner of making a proportional section. A threefold rate followeth: The first is of the greater segment. The converse is apparent in the same example: For seeing that i o, is of power five times so much as is a o; the gnomon l m n, shall be four times so much as is u a: Whose quadruple also, by the 14. e xij, is a v. Therefore it is equal to the gnomon. Now a j, is equal to a e: Therefore it is the double also of a o, that is of a y: And therefore by the 24. e x. it is the double of a t: And therefore it is equal to the compliments i y, and y s: Therefore the other diagonal y r, is equal to the other rectangle i v. Wherefore, by the 8 e xij. as e v, that is, a e, is to that t, that is a i: so is a i, unto i e; Wherefore by the ● e, a e, is proportional cut: And the greater segment is a i, the same remain. The other propriety of the quintuple doth follow. 6 The lesser segment continued to the half of the greater, is of power quintuple to the same half è 3 p x iij. The rate of the triple followeth. 7 The whole line and the lesser segment are in power treble unto the greater. è 4 p xiij. 8 An obliquangled parallelogramme is either a Rhombus, or a Rhomboides. 9 A Rhombus is an obliquangled equilater parallelogramme 32 dj. It is otherwise of some called a Diamond. 10 A Rhomboides is an obliquangled parallelogram●e not equilater 33. dj. And a Rhomboides is so opposed to an oblong, as a Rhombus is to a quadrate. And the Rhomboides is so called as one would say Rhombuslike, although beside the inequality of the angel's it hath nothing like to a Rhombus. An example of measuring of a Rhombus is thus. 11 A Trapezium is a quadrangle not parallelogramme. 34. dj. The examples both of the figure and of the measure of the same let these be. Therefore triangulate quadrangles are of this sort. 12 A multangle is a figure that is comprehended of more than four right lines. 23. dj. By this general name, all other sorts of right lined figures hereafter following, are by Euclid comprehended, as are the quinquangle, sexangle, septangle, and such like innumerable taking their names of the number of their angles. In every kind of multangle, there is one ordinate, as we have in the former signified, of which in this place we will say nothing, but this one thing of the quinquangle. The rest shall be reserved until we come to Adscription. 13 Multangled triangulates do take their measure also from their triangles. 14 If an equilater quinquangle have three sides equal, it is equiangled. 7 p 13. This of some, from the Greek is called a Pentagon; of others a Pentangle, by a name partly Greek partly Latin. The fifteenth Book of Geometry, Of the Lines in a Circle. AS yet we have had the Geometry of rectilineals: The Geometry of Curvilineals, of which the Circle is the chief, doth follow. 1. A Circle is a round plain. ● 15 dj. The means to describe a Circle, is the same, which was to make a Periphery: But with some difference: For there was considered no more but the motion, the point in the end of the ray describing the periphery: Here is considered the motion of the whole ray, making the whole plot contained within the periphery. A Circle of all plains is the most ordinate figure, as was before taught at the 10 e iiij. 2 Circle's are as the quadrates or squares made of their diameters 2 p. x ij. Therefore 3. The Diameters are, as their peripheries Pappus, 5 l x j, and 26 th'. 18. As here thou seest in a e, and i o. 4. Circular Geometry is either in Lines, or in the segments of a Circle. This partition of the subject matters howsoever is taken for the distinguishing and severing with some light a matter somewhat confused; And indeed concerning lines, the consideration of secants is here the foremost, and first of Inscripts. 5. If a right line be bounded by two points in the periphery, it shall fall within the Circle. 2 p iij. From hence doth follow the Infinite section, of which we spoke at the 6 e i. This proposition teacheth how a Rightline is to be inscribed in a circle, to wit, by taking of two points in the periphery. 6. If from the end of the diameter, and with a ray of it equal to the right line given, a periphery be described, a right line drawn from the said end, unto the meeting of the peripheries, shall be inscribed into the circle, equal to the right line given. 1 p iiij. And this proposition teacheth, How a right line given is to be inscribed into a Circle, equal to a line given. Moreover of all inscripts the diameter is the chief: For it showeth the centre, and also the reason or proportion of all other inscripts. Therefore the invention and making of the diameter of a Circle is first to be taught. 7. If an inscript do cut into two equal parts, another inscript perpendicularly, it is the diameter of the Circle, and the midst of it is the centre. 1 p iij. The cause is the same, which was of the 5 e x i. Because the inscript cut into halves if for the side of the inscribed rectangle, and it doth subtend the periphery cut also into two parts; By the which both the Inscript and Periphery also were in like manner cut into two equal parts: Therefore the right line thus halfing in the diameter of the rectangle: But that the middle of the circle is the centre, is manifest out of the 7 e v, and 29 e iiij. Euclid, thought better of Impossibile, than he did of the cause: And thus he forceth it. For if y be not the Centre, but s, the part must be equal to the whole: For the Triangle a o s, shall be equilater to the triangle e o s. For a o, oe, are equal by the grant: Item s a, and s e, are the rays of the circle: And s o, is common to both the triangles. Therefore by the 1 e seven, the angel's no each side at o are equal; And by the 13 e v, they are both right angles. Therefore s o e is a right angle; It is therefore equal by the grant, to the right angle y o e, that is, the part is equal to the whole, which is impossible. Wherefore y is not the Centre. The same will fall out of any other points whatsoever ●ut of y. Therefore 8. If two r●ght lines do perpendicularly half two inscripts, the meeting of these two bisecants shall be the Centre of the circle è 25 p iij. And one may 9 Draw a periphery by three points, which do not fall in a right line. 10. If a diameter do half an inscript, that is, n●t a diameter, it doth cut it perpendicularly: And chose: 3 p iij. 11. If inscripts which are not diameters do cut one another, the segments shall be unequal. 4 p iij. But rate hath been hitherto in the parts of inscripts: Proportion in the same parts followeth. 12 If two inscripts do cut one another, the rectangle of the segments of the one is equal to the rectangle of the segments of the other. 35 p iij. And this is the comparison of the parts inscripts. The rate of whole inscripts doth follow, the which whole one diameter doth make: 13 Inscripts are equal distant from the centre, unto which the perpendiculars from the centre are equal 4 d iij. 14. If inscripts be equal, they be equally distant from the centre: And chose. 13 p iij. The diameters in the same circle, by the 28 e iiij● are equal: And they are equally distant from the centre, seeing they are by the centre, or rather are no whit at all distant from it: Other inscripts are judged to be equal, greater, or lesser one than another, by the diameter, or by the diameters centre. Euclid doth demonstrate this proposition thus: Let first a e and i o be equal; I say they are equidistant from the centre. For let u y, and u y, be perpendiculars: They shall cut the assigned a e, & i o, into halves, by the 5 e xj: And y a and s i a●e equal, because they are the halves of equals. Now let the rays of the circle be u a, aund u i: Their quadrates by the 9 e xij, are equal to the pair of quadrates of the shanks, which pairs are therefore equal between themselves. Take from equals the quadrates y a, and s i, there shall remain y u and u s, equals: and therefore the sides are equal, by the 4 e 12. The converse likewise is manifest: For the perpendiculars given do half them: And the halves as before are equal. 15 Of unequal inscripts the diameter is the greatest: And that which is next to the diameter, is greater than that which is farther off from it: That which is farthest off from it, is the least: And that which is next to the least, is lesser than that which is farther off: And those two only which are on each side of the diameter are equal è 15 e iij. This proposition consisteth of five members: The first is, The diameter is the greatest iuscript: The second, That which is next to the diameter is greater than that which is farther off: The third, That which is farthest off from the diameter is the least: The fourth, That next to the least is lesser, than that farther off: The fifth, That two only on each side of the diameter are equal between themselves. All which are manifest out of that same argument of equality, that is the centre the beginning of decreasing, and the end of increasing. For look how much farther off you go from the centre, or how much nearer you come unto it, so much les●er or greater do you make the inscript. But Euclides conclusion is by triangles of two sides greater than the other; and of the greater angle. The first part is plain thus: Because the diameter a e, is equal to i l, and l o, viz. to the rays: And to those which are greater than i o, the base by the 9 e v j etc. The second part of the nearer, is manifest by the 5 e seven. because of the triangle i l oh, equicrural to the triangle u l y, is greater in angle: And therefore it is also greater in base. The third and fourth are consectaries of the first. The fifth part is manifest by the second: For if beside i o, and s r, there be supposed a third equal, the same also shall be unequal, because it shall be both nearer and farther off from the diameter. 16 Of right lines drawn from a point in the diameter which is not the centre unto the periphery, that which passeth by the centre is the greatest: And that which is nearer to the greatest, is greater than that which is farther off: The other part of the greatest is the jest. And that which is nearest to the least, is lesser than that which is farther off: And two on each side of the greater or least are only equal. 7 p iij. The third, that a y, is lesser than a u, because s y, which is equal to s u, is lesser than the right lines s a, and a u, by the 9 e v j: And the common s a, being taken away, a y shall be left, lesser than a u. The fourth part followeth of the third. The fifth let it be thus: s r, making the angle a s r, equal to the angle a s u, the bases a u, and a r, shall be equal by the 2 e v ij. To these if the third be supposed to be equal, as a l, it would follow by the 1 e v ij. that the whole angle s a, should be equal to r s a, the particular angle, which is impossible. And out of this fifth part issueth this Consectary. Therefore 17 If a point in a circle be the bound of three equal right lines determined in the periphery, it is the centre of the circle. 9 p iij. Let the point a, in a circle be the common bound of three right lines, ending in the periphery and equal between themselves, be a e, a i, a u I say this point is the centre of the Circle. 18 Of right lines drawn from a point assigned without the periphery, unto the concavity or hollow of the same, that which is by the centre is the greatest; And that next to the greatest, is greater than that which is farther off: But of those which fall upon the convexiti● of the circumference, the segment of the greatest is least● And that which is next unto the least is lesser than that is farther off: And two on each side of the greatest or least are only equal. 8 piij. 19 If a right line be perpendicular unto the end of the diameter, it doth touch the periphery: And chose è 16 p iij. As for example, Let the circle given a e, be perpendicular to the end of the diameter, or the end of the ray, in the end a, as suppose the ray be i a: I say, that e a, doth touch, not cut the periphery in the common bound a. Therefore 20 If a right line do pass by the centre and touch-point, it is perpendicular to the tangent or touch-line. 18 p iij. And Or thus, as Schoner amendeth it: If a right line be the diameter by the touch point, it is perpendicular to the tangent. 21 If a right line be perpendicular unto the tangent, it doth pass by the centre and touch-point. 19 piij. Or thus: if it be perpendicular to the tangent, it is a diameter by the touch point: Schoner. For a right line either from the centre unto the touch-point; or from the touch point unto the centre is radius or semidiameter. And 22 The touch-point is that, into which the perpendicular from the centre doth fall upon the touch line. 23 A tangent on the same side is only one. Or touch line is but one upon one, and the same side: H. Or. A tangent is but one only in that point of the periphery Schoner. Euclid propoundeth this more specially thus; that no other right line may possibly fall between the periphery and the tangent. And 24 A touch-angle is lesser than any rectilineall a●ute angle, è 16 p ij. Angulus contractus, A touch angle is an angle of a strait touch-line and a periphery. It is commonly called Angulus contingentiae: Of Proclus it is named Cornicularis, an horne-like corners because it is made of a right line and periphery like unto a horn. It is less therefore than any acute or sharp rightlined angle: Because if it were not lesser, a right line might fall between the periphery and the perpendicular. And 25 All touch-angles in equal peripheries are equal. But in unequal peripheries, the cornicular angle of a lesser periphery, is greater than the Cornicular of a greater. 26 If from a ray out of the centre of a periphery given, a periphery be described unto a point assigned without, and from the meeting of the assigned and the ray, a perpendicular falling upon the said ray unto the now described periphery, be tied by a right line with the said centre, a right line drawn from the point given unto the meeting of the periphery given, and the knitting line shall touch the assigned periphery 17 p iij. Thus much of the Secants and Tangents severally: It followeth of both kinds jointly together. 27 If of two right lines, from an assigned point without, the first do cut a periphery unto the concave, the other do touch the same; the oblong of the secant, and of the outer segment of the secant, is equal to the quadrate of the tangent: and if such a like oblong be equal to the quadrate of the other, that same other doth touch the periphery: 36 , and 37 . p iij. Therefore 28. All tangents falling from the same point are equal. Or, Touch lines drawn from one and the same point are equal: H. Because their quadrates are equal to the same oblong. And 29. The oblongs made of any secant from the same point, and of the outer segment of the secant are equal between themselves. Camp. 36 p iij. The reason is because to the same thing. And 30. To two right lines given one may so continue or join the third, that the oblong of the continued and the continuation may be equal to the quadrate remaining. Vitellio 127 p i. As in the first figure, if the first of the lines given be e o, the second i a, the third o a. Now are we come to Circular Geometry, that is to the Geometry of Circles or Peripheries cut and touching one another: And of Right lines and Peripheries. 31. If peripheries do either cut or touch one another, they are eccentrickes: And they do cut one another in two points only, and these by the touch point do continue their diameters, 5. 6. 10,11, 12 p iij. All these might well have been asked: But they have also their demonstrations, ex impossibili, not very dissicult. Of right lines and Peripheries jointly the rate is but one. 32. If inscripts be equal, they do cut equal peripheries: And chose, 28,29 p iij. Or thus: If the inscripts of the same circle or of equal circles be equal, they do cut equal peripheries: And chose B. Or thus; If lines inscribed into equal circles or to the same be equal, they cut equal peripheries: And chose, if they do cut equal peripheries, they shall themselves be equal: Schoner● Except with the learned Rodulphus Snellius, you do understand aswell two equal peripheries to be given, as two equal right lines, you shall not conclude two equal sections, and therefore we have justly inserted of the same, or of equal Circles; which we do now see was in like manner by Lazarus Schonerus. The sixteenth Book of Geometry, Of the Segments of a Circle. 1. A Segment of a Circle is that which is comprehended outterly of a periphery, sand innerly of a r●ght line. THe Geometry of Segments is common also to the sphere: But now this same general is hard to be declared and taught: And the segment may be comprehended within of an oblique line either single or manifold. But here we follow those things that are usual and commonly received. First therefore the general definition is set foremost, for the more easy distinguishing of the species and several kinds. 2. A segment of a Circle is either a sectour, or a s●ction. Segmentum a segment, and Sectio a section, and Sector a sectour, are almost the same in common acceptation, but they shall be distinguished by their definitions. 3. A Sectour is a segment innerly comprehended of two right lines, making an angle in the centre; which is called an angle in the centre: As the periphery is, the base of the sectour, 9 d iij. 4. An angle in the Periphery is an angle comprehended of two right lines inscribed, and jointly bounded or meeting in the periphery. 8 d iij. This might have been called The Sectour in the periphery, to wit, comprehended innerly of two right lines jointly bounded in the periphery; as here a e i. 5. The angle in the centre, is double to the angle of the periphery standing upon the same base, 20 p iij. Therefore 6. If the angle in the periphery be squall to the angle in the centre, it is double to it in base. And chose. This followeth out of the former element: For the angle in the centre is double to the angle in the periphery standing upon the same base: Wherefore if the angle in the periphery be to be made equal to the angle in the centre, his base is to be doubled, and thence shall follow the equality of them both: S. 7. The angles in the centre or periphery of equal circles, are as the Peripheries are upon which they do insist: And chose. è 33 p uj, and 26, 27 p iij. Here is a double proportion with the periphery underneath, of the angles in the centre: And of angles in the periphery. But it shall suffice to declare it in the angles in the centre. First therefore let the Angles in the centre a e i, and o u y, be equal: The bases a i, and oh y, shall be equal, by the 11 e seven: And the peripheries a i, and oh y, by the 32 e x v, shall likewise be equal. Therefore if the angles be unequal, the peripheries likewise shall be equal. The same shall also be true of the Angles in the Periphery. The Converse in like manner is true: From whence followeth this consectary: Therefore 8. As the sectour is unto the sectour, so is the angle unto the angle: And chose. And thus much of the Sectour. 9 A section is a segment of a circle within comprehended of one right line, which is termed the base of the section. As here, a e i, and o u y, and s r l, are sections. 10. A section is made up by finding of the centre. 11 The periphery of a section is divided into two equal parts by a perpendicular dividing the base into two equal parts. 20. p iij. Here Euclid doth by congruency comprehend two peripheries in one, and so do we comprehend them. 12 An angle in a section is an angle comprehended of two right lines jointly bounded in the base and in the periphery jointly bounded 7 d iij. Or thus: An angle in the section, is an angle comprehended under two right lines, having the same terms with the bases, and the terms with the circumference: H. As a o e, in the former example. 13 The angles in the same section are equal. 21. p iij. Here it is certain that angles in a section are indeed angel's in a periphery, and do differ only in base. 14 The angles in opposite sections are equal to two right angles. 22. p iij. The reason or rate of a section is thus: The similitude doth follow. 15 If sections do receive [or contain] equal angles, they are alike e 10. d iij. 16 If like sections be upon an equal base, they are equal: and chose. 23,24. p iij. In the first figure, let the base be the same. And if they shall be said to unequal sections; and one of them greater than another, the angle in that a o e, shall be less than the angle a i e, in the lesser section, by the 16 e vi. which notwithstanding, by the grant, is equal. In the second figure, if one section be put upon another, it will agree with it: Otherwise against the first part, like sections upon the same base, should not be equal. But congruency is here sufficient. By the former two propositions, and by the 9 e x v. one may find a section like unto another assigned, or else from a circle given to cut off one like unto it. 17 An angle of a section is that which is comprehended of the bounds of a section. 18 A section is either a semicircle: or that which is unequal to a semicircle. A section is two fold, a semicircle, to wit, when it is cut by the diameter: or unequal to a semicircle, when it is cut by a line lesser than the diameter. 19 A semicircle is the half section of a circle. Or it is that which is made the diameter. Therefore 20 A semicircle is comprehended of a periphery and the diameter 18 dj. 21 The angle in a semicircle is a right angle: The angle of a semicircle is lesser than a rectilineall right angle: But greater than any acute angle: The angle in a greater section is lesser than a right angle: Of a greater, it is a greater. In a lesser it is greater: Of a lesser, it is lesser, ê 31 . and 16. p iij. Or thus: The angle in a semicircle is a right angle, the angle of a semicircle is less than a right rightlined angle, but greater than any acute angle: The angle in the greater section is less than a right angle: the angle of the greater section is greater than a right angle: the angle in the lesser section is greater than a right angle, the angle of the lesser section, is lesser than a right angle: H. The second part, That the angle of a semicircle is lesser than a right angle; is manifest out of that, because it is the part of a right angle. For the angle of the semicircle a i e, is a part of the rectilineall right angle a i u. The third part, That it is greater than any acute angle; is manifest out of the 23. e x v. For otherwise a tangent were not on the same part one only and no more. The fourth part is thus made manifest: The angle at i, in the greater section a e i, is lesser than a right angle; because it is in the same triangle a e i, which at a, is right angle. And if neither of the shanks be by the centre, notwithstanding an angle may be made equal to the assigned in the same section. The fifth is thus: The angle of the greater section e a i, is greater than a right angle; because it containeth a rightangle. The sixth is thus, the angle a oh e, in a lesser section, is greater than a right angle, by the 14 e x v i. Because that which is in the opposite section, is lesser than a right angle. The seventh is thus. The angle e a o, is lesser than a rightangle: Because it is part of a right angle, to wit of the outer angle, if i a, be drawn out at length And thus much of the angles of a circle, of all which the most effectual and of greater power and use is the angle in a semicircle, and therefore it is not without cause so often mentioned of Aristotle. This Geometry therefore of Aristotle, let us somewhat more fully open and declare. For from hence do arise many things. Therefore 22 If two right lines jointly bounded with the diameter of a circle, be jointly bounded in the periphery, they do make a right angle. Or thus: If two right lines, having the same terms with the diameter, be joined together in one point, of the circomference, they make a right angle. H. This corollary is drawn out of the first part of the former Element, where it was said, that an angle in a semicircle is a right angle. And 23 If an infinite right line be cut of a periphery of an external centre, in a point assigned and contingent, and the diameter be drawn from the contingent point, a right line from the point assigned knitting it with the diameter, shall be perpendicular unto the infinite line given. Let the infinite right line be a e, from whose point a, a perpendicular is to be raised. And 24 If a right line from a point given, making an acute angle with an infinite line, be made the diameter of a periphery cutting the infinite, a right line from the point assigned knitting the segment, shall be perpendicular upon the infinite line. As in the same example, having an external point given, let a perpendicular unto the infinite right line a e, be sought: Let the right line i oh e, be made the diameter of the periphery; and withal let it make with the infinite right line giyen an acute angle in e, from whose bisection for the centre, let a periphery cut the infinite, etc. And 25 If of two right lines, the greater be made the diameter of a circle, and the lesser jointly bounded with the greater and inscribed, be knit together, the power of the greater shall be more than the power of the lesser by the quadrate of that which knitteth them both together. ad 13 p. x. 26 If a right line continued or continually made of two right lines given, be made the diameter of a circle, the perpendicular from the point of their continuation unto the periphery, shall be the mean proportional between the two lines given. 13 p vi. So if the side of a quadrate of 10. foot content, were sought; let the sides 1, foot and 10, foot an oblong equal to that same quadrate, be continued; the mean proportional shall be the side of the quadrate, that is, the power of it shall be 10. foot. The reason of the angles in opposite sections doth follow. 27 The angles in opposite sections are equal in the alterne angles made of the secant and touch line. 32. p iij. As let the unequal sections be e i o, and e a o: the tangent let it be u e y: And the angles in the opposite sections, e a o, and e i o. I say they are equal in the alterne angles of the secant and touch line o e y, and o e u. First that which is at a, is equal to the alterne o e y: Because also three angles o e y, oh e a, and a e u, are equal to two right angles, by the 14 e v. Unto which also are equal the three angles in the triangle a e o, by the 13 e vi. From three equals take away the two right angles a u e, and a o e: (For a o e, is a right angle, by the 21 e; because it is in a semicircle:) Take away also the common angle a e o: And the remainders e a o, and o e y, alterne angles, shall be equal. Therefore 28 If at the end of a right line given a right lined angle be made equal to an angle given, and from the top of the angle now made, a perpendicular unto the other side do meet with a perpendicular drawn from the midst of the line given, the meeting shall be the centre of the circle described by the equalled angle, in whose opposite section the angle upon the line given shall be made equal to the assigned è 33 p iij. And 29 If the angle of the secant and touch line be equal to an assigned rectilineall angle, the angle in the opposite section shall likewise be equal to the same. 34. piij. Of Geometry the seventeenth Book, Of the Adscription of a Circle and Triangle. HItherto we have spoken of the Geometry of Rectilineall plains, and of a circle: Now followeth the Adscription of both: This was generally defined in the first book 12 e. Now the periphery of a circle is the bound thereof. Therefore a rectilineall is inscribed into a circle, when the periphery doth touch the angles of it 3 d iiij. It is circumscribed when it is touched of every side by the periphery; 4 d iij. 1. If a rectilineall ascribed unto a circle be an equilater, it is equiangle. Of the circumscript it is likewise true, if the circumscript be understood to be a circle. For the perpendiculars from the centre a, unto the sides of the circumscript, by the 9e xij, shall make triangles on each side equilaters, & equiangls, by drawing the semidiameters unto the corners, as in the same example. 2. It is equal to a triangle of equal base to the perimeter, but of height to the perpendicular from the centre to the side. As here is manifest, by the 8 e seven. For there are in one triangle, three triangles of equal height. The same will fall out in a Triangulate, as here in a quadrate: For here shall be made four triangles of equal height. Lastly every equilater rectilineall ascribed to a circle, shall be equal to a triangle, of base equal to the perimeter of the adscript. Because the perimeter containeth the bases of the triangles, into the which the rectilineall is resolved. 3. Like rectilinealls inscribed into circles, are one to another as the quadrates of their diameters, 1 p. x i i. In like Triangulates, seeing by the 4 e x, they may be resolved into like triangles, the same will fall out. Therefore 4. If it be as the diameter of the circle is unto the side of rectilineall inscribed, so the diameter of the second circle be unto the side of the second rectilineall inscribed, and the several triangles of the inscripts be alike and likely situate, the rectilinealls inscribed shall be alike and likely situate. This Euclid did thus assume at the 2 p xij, and indeed as it seemeth out of the 18 p vi. Both which are contained in the 23 e iiij. And therefore we also have assumed it. Adscription of a Circle is with any triangle: But with a triangulate it is with that only which is ordinate: And indeed adscription of a Circle is common to all. 5. If two right lines do cut into two equal parts two angles of an assigned rectilineall, the circle of the ray from their meeting perpendicular unto the side, shall be inscribed unto the assigned rectilineall. 4 and 8. p. iiij. The same argument shall serve in a Triangulate. 6. If two right lines do right anglewise cut into two equal parts two sides of an assigned rectilineall, the circle of the ray from their meeting unto the angle, shall be circumscribed unto the assigned rectilineall. 5 p iiij. As in the former figures. The demonstration is the same with the former. For the three rays, by the 2 e seven, are equal: And the meeting of them, by the 17 ex, is the centre. And thus is the common adscription of a circle: The adscription of a rectilineall followeth, and first of a Triangle. 7. If two inscripts, from the touch point of a right line and a periphery, do make two angles on each side equal to two angles of the triangle assigned be knit together, they shall inscribe a triangle into the circle given, equiangular to the triangle given è 2 p iiij. The circumscription here is also special. 8 If two angles in the centre of a circle given, be equal at a common ray to the outer angel's of a triangle given, right lines touching a periphery in the shanks of the angles, shall circumscribe a triangle about the circle given like to the triangle given. 3 p iiij. Therefore 9 If a triangle be a rectangle, an obtusangle, an acute angle, the centre of the circumscribed triangle is in the side, out of the sides, and within the sides: And chose. 5 e iiij. As, thou seest in these three figures underneath, the centre a. Of Geometry, the eighteenth Book, Of the adscription of a Triangulate. Such is the Adscription of a triangle: The adscription of an ordinate triangulate is now to be taught. And first the common adscription, and yet out of the former adscription, after this manner. 1. If right lines do touch a periphery in the angles of the inscript ordinate triangulate, they shall unto a circle circumscribe a triangulate homogeneal to the inscribed triangulate. The examples shall be laid down according as the species or several kinds do come in order. The special inscription therefore shall first be taught, and that by one side, which reiterated, as oft as need shall require, may fill up the whole periphery. For that Euclid did in the quindecangle one of the kinds, we will do it in all the rest. 2. If the diameters do cut one another right-anglewise, a right line subtended or drawn against the right angle, shall be the side of the quadrate. è 6 p iiij. Therefore 3. A quadrate inscribed is the half of that which is circumscribed. Because the side of the circumscribed (which here is equal to the diameter of the circle) is of power double, to the side of the inscript, by the 9 e x i i. An● 4. It is greater than the half of the circumscribed Circle. Because the circumscribed quadrate, which is his double, is greater than the whole circle. For the inscribing of other multangled odde-sided figures we must needs use the help of a triangle, each of whose angles at the base is manifold to the other: In a Quinguangle first, that which is double unto the remainder, which is thus found. 5. If a right line be cut proportionally, the base of that triangle whose shanks shall be equal to the whole line cut, and the base to the greater segment of the same, shall have each of the angles at the base double to the remainder: And the base shall be the side of the quinquangle inscribed with the triangle into a circle. 10, and 11. p i i i i. 6 If two right lines do subtend on each side two angles of an inscript quinquangle, they are cut proportionally, and the greater segments are the sides of the said inscript è 8, p x iij. And from hence the fabric or construction of an ordinate quinquangle upon a right line given, is manifest. Therefore 7 If a right line given, cut proportional, be continued at each end with the greater segment, and six peripheries at the distance of the line given shall meet, two on each side from the ends of the line given and the continued, two others from their meetings, right lines drawn from their meetings, & the ends of the assigned shall make an ordinate quinquangle upon the assigned. 8 If the diameter of a circle circumscribed about a quinquangle be rational, it is irrational unto the side of the inscribed quinquangle. è 11. p xiij. So before the segments of a right line proportionally cut were irrational. The other triangulates hereafter multiplied from the ternary, quaternary, or quinary of the sides, may be inscribed into a circle by an inscript triangle, quadrate, or quinquangle. Therefore by a triangle there may be inscribed a triangulate of 6. 12,24,46, angles: By a quadrate, a triangulate of 8. 16,32,64. angles. By a quinquangle, a triangulate of 10, 20. 40,80. angles etc. 9 The ray of a circle is the side of the inscript sexangle è 15 p iiij. Therefore 10 Three ordinate sexangles do fill up a place. Furthermore also no one figure amongst the plains doth fill up a place. A Quinquangle doth not: For three angles a quinquangle may make only 3 ●/5 angles which is too little. And four would make 4 ●/5. which is as much too great. The angles of a septangle would make only two rightangles, and 6/7 of one: Three would make 3, and 9/7, that is in the whole 4. 2/7, which is too much, etc. to him that by induction shall thus make trial, it will appear, That a plain place may be filled up by three sorts of ordinate plains only. And 11 If right lines from one angle of an inscript sexangle unto the third angle on each side be knit together, they shall inscribe an equilater triangle into the circle given. 12 The side of an inscribed equilater triangle hath a treble power, unto the ray of the circle 12. p xiij. 13 If the side of a sexangle be cut proportionally, the greater segment shall be the side of the decangle. Therefore 14 If a decangle and a sexangle be inscribed in the same circle, a right line continued and made of both sides, shall be cut proportionally, and the greater segment shall be the side of a sexangle; and if the greater segment of a right line cut proportionally be the side of an hexagon, the rest shall be the side of a decagon. 9 p xiij. The comparison of the decangle and sexangle with the quinangle followeth. 15 If a decangle, a sexangle, and a pentangle be inscribed into the same circle the side of the pentangle shall in power countervail the sides of the others. And if a right line inscribed do countervail the sides of the sexangle and decangle, it is the side of the pentangle. 10. p xiv. Let the proportion of this syllogism be demonstrated: For this part only remaineth doubtful. Therefore two triangles a e i, and the e i, are equiangles, having one common angle at e: And also two equal ones a e i, and e i y, the halves, to wit, of the same e i s: Because that is, by the 17 e, uj: one of the two equals, unto the which e ay s, the out angle, is equal, by the 15 e. vi. And this doth insist upon a half periphery. For the half periphery a l s, is equal to the half periphery a r s: and also a l, is equal to a r. Therefore the remnant l s, is equal to the remnant r s: And the whole r l, is the double of the same r s: And therefore e r, is the double of e o: And r s, the double of o u. For the bisegments are manifest by the 10 e, xv. and the 11 e, xuj. Therefore the periphery e r s, is the double of the periphery e o u: And therefore the angle e i u, is the half of the angle e i s, by the 7 e, xuj. Therefore two angles of two triangles are equal: Wherefore the remainder, by the 4 e seven, is equal to the remainder. Wherefore by the 12 e, seven, as the side a e, is to e i: so is e i, to e y. Therefore by the 8 e xij, the oblong of the extremes is equal to the quadrate of the mean. Now let o y, be knit together with a strait: Here again the two triangles a o e, and a o y, are equiangles, having one common angle at a: And a oh y, and oh e a, therefore also equal: Because both are equal to the angle at a: That by the 17 e, uj: This by the 2 e, seven: Because the perpendicular halfing the side of the decangle, doth make two triangles, equicrural, and equal by the right angle of their shanks: And therefore they are equiangles. Therefore as e a, is to a o: so is e a, to a y. Wherefore by the 8 e, xij. the oblong of the two extremes is equal to the quadrate of the mean: And the proposition of the syllogism, which was to be demonstrated. The converse from hence as manifest Euclid doth use at the 16 p xiij. 16. If a triangle and a quinquangle be inscribed into the same Circle at the same point, the right line inscribed between the bases of the both opposite to the said point, shall be the side of the inscribed quindecangle. 16. p. iiij. Therefore 17. If a quinquangle and a sexangle be inscribed into the same circle at the same point, the periphery intercepted between both their sides, shall be the thirtieth part of the whole periphery. Of Geometry the ninteenth Book; Of the Measuring of ordinate Multangle and of a Circle. Out of the Adscription of a Circle and a Rectilineall is drawn the Geodesy of ordinate Multangles, and first of the Circle itself. For the meeting of two right lines equally, dividing two angles is the centre of the circumscribed Circle: From the centre unto the angle is the ray: And then if the quadrate of half the side be taken out of the quadrate of the ray, the side of the remainder shall be the perpendicular, by the 9 e xij. Therefore a special theorem is here thus make: 1. A plain made of the perpendicular from the centre unto the side, and of half the perimeter, is the content of an ordinate multangle. In an ordinate Sexangle also the ray, by the 9 e xviij, is known by the side of the sexangled As here, the quadrate of 6, the ray is 36. The quadrate of 3, the half of the side, is 9● And 36— 9 are 27, for the quadrate of the Perpendicular, whose side 5 12/11 is the perpendicular itself. Now the whole perimeter, as you see, is 36. Therefore the half is 18. And the product of 18 by 5 2/11 is 93 3/11 for the content of the sexangle given. Lastly in all ordinate Multangles this theorem shall satisfy thee. 2 The periphery is the triple of the diameter and almost one seav●nth part of it. Therefore 3. The plain of the ray, and of half the periphery is the content of the circle. And 4. As 14 is unto 11, so is the quadrate of the diameter unto the Circle. For here 3 bounds of the proportion are given in potentia: The fourth is found by the multiplication of the third by the second, and by the Division of the product by the first: As here the Quadrate of the diameter 14, is 196. The product of 196 by 11 is 2156. Lastly 2156 divided by 14, the first bound, giveth in the Quotient 154, for the content of the circle sought. This ariseth by an analysis out of the quadrate and Circle measured. For the reason of 196, unto a 154; is the reason of 14 unto 11, as will appear by the reduction of the bounds. This is the second manner of squaring of a circle taught by Euclid as Hero telleth us, but otherwise laid down, namely after this manner. If from the quadrate of the diameter you shall take away 3/14 parts of the same, the remainder shall be the content of the Circle. As if 196, the quadrate be divided by 14, the quotient likewise shall be 14. Now thrice 14, are 42: And 196— 42, are 154, the quadrate equal to the circle. Out of that same reason or rate of the pheriphery and diameter ariseth the manner of measuring of the Parts of a circle, as of a Semicircle, a Sector, a Section, both greater and lesser. And 5. The plain of the ray and one quarter of the periphery, is the content of the semicircle. This may also be done by taking of the half of the circle now measured. And 6. The plain made of the ray and half the base, is the content of the Sector. And 7. If a triangle, made of two rays and the base of the greater section, be added unto the two sectors in it, the whole shall be the content of the greater section: If the same be taken from his own sector, the remainder shall be the content of the lesser. In the former figure the greater section is a e i: The lesser is a i. The base of them both is as you see, 6. The perpendicular from the top of the triangle, or his height is 4. Therefore the content of the triangle is 12. Wherefore 30— 30— 12, that is 72, is the content of the greater section a e i. And the lesser sectour, as in the former was taught, is 184/7. Therefore 184/7— 12, that is, 6 4/7, is the content of a i, the lesser section. And 8. A circle of unequal isoperimetrall plains is the greatest. Of Geometry the twentieth Book, Of a Bossed surface. 1. A bossed surface is a surface which lieth unequally between his bounds. IT is contrary unto a Plain surface, as we heard at the 4 e v. 2. A bossed surface is either a spherical, or varium. Therefore 4 It is made by the turning about of an half circumference the diameter standeth still. è 14 d x i. As here if thou shalt conceive the space between the periphery and the diameter to be empty. 5. The greatest periphery in a spherical surface is that which cutteth it into two equal parts. Those things which were before spoken of a circle, the same almost are hither to be referred. The greatest periphery of a spherical doth answer unto the Diameter of a Circle. Therefore 6. That periphery that is nearer to the greatest, is greater than that which is farther off: And on each side those two which are equally distant from the greatest, are equal. 7 The plain made of the greatest periphery and his diameter is the spherical. Therefore 8 A plain of the greatest circle and 4, is the spherical. This consectarium is manifest out of the former element. And 9 As 7 is to 22. so is the quadrate of the diameter unto the spherical. For 7, and 22, are the two least bounds in the reason of the diameter unto the periphery: But in a circle, as 14, is to 11. so is the quadrate of the diameter unto the circle. The analogy doth answer fitly: Because here thou multipliest by the double, and dividest by the half: There chose thou multipliest by the half, and dividest by the double. Therefore there one single circle is made, here the quadruple of that. This is, therefore the analogy of a circle and spherical; from whence ariseth the hemispherical, the greater and the lesser section. And 10 The plain of the greatest periphery and the ray, is the hemisphericall. As here, the greatest periphery is 44. the ray 7. The product therefore of 44. by 7. that is, 308. is the hemisphericall. 11 If look what the part be of the ray perpendicular from the centre unto the base of the greater section, so much the hemisphericall be increased, the whole shall be the greater section of the spherical: But if it be so much decreased, the remainder shall be the lesser. 12 The varium is a bossed surface, whose base is a periphery, the side a right line from the bound of the top, unto the bound of the base. 13 A varium is a conical or a cylinder like form. Therefore 15 It is made by turning about of the side about the periphery beneath. 16 The plain of the side and half the base is the conical surface. As in the example next afore going, the side is 13. The half periphery is 15 5/7: And the product of 15. 5/7. by 13. is 204. 2/7. for the conical surface. To which if you shall add the circle underneath, you shall have the whole surface. Therefore 18 It is made by the turning of the side about two equal and parallel peripheries. 19 The plain of his side and height is the cylinder like surface. As here the periphery is 22. as is gathered by the Diameter, which is 7. The height is 12. The base therefore is 38. ½. And 38. ½ by 12. are 462. for the cylinderlike surface. To which if you shall add both the bases on each side, to wit, 38. ½. twice, or 77. once, the whole surface shall be 539. Geometry, the one and twentieth Book. Of Lines and Surfaces in solidest 1 A body or solid is a lineate broad and high 1 d xj. 2 The bound of a solid is a surface 2 d xj. The bound of a line is a point: and yet neither is a point a line, or any part of a line. The bound of a surface is a line: And yet a line is not a surface, or any part of a surface. So now the bound of a body is a surface: And yet a surface is not a body, or any part of a body. A magnitude is one thing; a bound of a magnitude is another thing, as appeared at the 5 e i. As they were called plain lines, which are conceived to be ●● a plain, so those are named solid both lines and surfaces which are considered in a solid; And their perpendicle and parallelisme are hither to be recalled from simple lines. 3 If a right line be unto right lines cut in a plain underneath, perpendicular in the common intersection, it is perependicular to the plain beneath: And if it be perpendicular, it is unto right lines, cut in the same plain, perpendicular in the common intersection è 3 d and 4 pxj. If thou shalt conceive the right lines, a e, i o, u y, to cut one another in the plain beneath, in the common intersections: And the line r s, falling from above, to be to every one of them perpendicular in the common point s, thou hast an example of this consectary. 4 If three right lines cutting one another, be unto the same right line perpendicular in the common section, they are in the same plain 5. p x i. For by the perpendicle and common section is understood an equal state on all parts, and therefore the same plain: as in the former example, a s, y s, o s, suppose them to be to s r, the same lofty line, perpendicular, they shall be in the same nearer plain a i u e o y. 5 If two right lines be perpendicular to the underplaine, they are parallels: And if the one of two parallels be perpendicular to the under plain, the other is also perpendicular to the same. 6.8 p xj. 6 If right lines in divers plains be unto the same right line parallel, they are also parallel between themselves. 9 p xj. 7 If two right lines be perpendiculars, the first from a point above, unto a right line underneath, the second from the common section in the plain ●nderneath, a third, from the said point perpendicular to the second, shall be perpendicular to the plain beneath. è 11 p xj. If the right line i o, do with equal angles agree to r, the third element. 8. If a right line from a point assigned of a plain underneath, be parallel to a right line perpendicular to the same plain, it shall also be perpendicular to the plain underneath. e x 12 p xj. 9 If a right line in one of the plains cut, perpendicular to the common section, be perpendicular to the other, the plains are perpendicular: And if the plains be perpendicular, a right line in the one perpendicular to the common section is perpendicular to the other è 4 d, and 38 p xj. 10. If a right line be perpendicular to a plain, all plains by it, are perpendicular to the same: And if two plains be unto any other plain perpendiculars, the common section is perpendicular to the same. e 15, and 19 p. xj. 11. Plains are parallel which do lean no way. 8 d x i. And 12. Those which divided by a common perpendicle. 14 p xj. It is also out of the definition of parallels, at the 17 e i i. And 13. If two pairs of right in them be jointly bounded, they are parallel. 15 p xj. The same will fall out if thou shalt imagine the jointly bounded to infinitely drawn out; for the plains also infinitely extended shall be parallelly 14. If two parallel plains are cut with another plain, the common sections are parallels, 16 p xj. The twenty second Book, of P. Ramus Geometry, Of a Pyramid. 1. The axis of a solid is the diameter about which it is turned, e 15,19,22 d x i. 2. A right solid is that whose axis is perpendicular to the centre of the base. Thus Serenus and Apollonius do define a Cone and a Cylinder: And these only Euclid considered: Yea and indeed stereometry entertaineth no other kind of solid but that which is right or perpendicular. 3. If solids be comprehended of homogeneal surfaces, equal in multitude and magnitude, they are equal. 10 d x i. Equality of lines and surfaces was not informed by any peculiar rule; farther than out of reason and common sense, and in most places congruency and application was enough and did satisfy to the full: But here the congruency of Bodies is judged by their surfaces. Two cubes are equal, whose six sides or plain surfaces, are equal, etc. 4. If solids be comprehended of surfaces in multitude equal and like, they are equal, 9 d x i. This is a consectary drawn out of the general definition of like figures, at the 19 e. iiij. For there like figures were defined to be equiangled and proportional in the shanks of the equal angles: But in like plain solids the angles are esteemed to be equal out of the similitude of their like plains: And the equal shanks are the same plain surfaces, and therefore they are proportional, equal and alike. 5 Like solids have a treble reason of their homologal sides, and two mean proportionals. 33. p xj. 8 p xij. It is a consectary drawn out of the 24 e. iiij. as the example from thence repeated shall make manifest. 6 A solid is plain or embosed. 7 A plain solid is that which is comprehended of plain surfaces. 8 The plain angles comprehending a solid angle, are less than four right angles. 21. p x j. For if they should be equal to four right angles, they would fill up a place, by the 22 e, vi. neither would they at all make an angle, much less therefore would they do it if they were greater. 9 If three plain angles less than four right angles, do comprehend a solid angle, any two of them are greater than the other● And if any two of them be greater than the other, then may comprehend a solid angle, 21. and 23. p xj. The converse from hence also is manifest. Euclid doth thus demonstrate it: First if three angles are equal, then by and by two are conceived to be greater than the remainder. But if they be unequal, let the angle a e i, be greater than the angle a e o: And let a e u, equal to a e o, be cut off from the greater a e i: And let e u, be equal to e o. Now by the 2 e, seven. two triangles a e u, and a e o, are equal in their bases a u, and a o. Item a o, and e i, are greater than a i, and a o: And a o, is equal to a u Therefore o ay, is greater than i u. Here two triangles, u e i, and i e o, equal in two shanks; and the base o ay, greater than the base i u. Therefore, by the 5 e seven. the angle oh e i, is greater than the angle i e u. Therefore two angles a e o, and o e i, are greater than a e i. 10 A plain solid is a Pyramid or a Pyramidate. 11 A Pyramid is a plain solid from a rectilineall base equally decreasing. As here thou conceivest from the triangular base a e i, unto the top o, the triangles a o e, a o ay, and e o ay, to be reared up. Therefore 12 The sides of a pyramid are one more than are the base. The sides are here named Hedrae. And 13 A pyramid is the first figure of solids. For a pyramid in solids, is as a triangle is in plains. For a pyramid may be resolved into other solid figures, but it cannot be resolved into any one more simple than itself, and which consists of fewer sides than it doth. Therefore 14 Pyramids of equal height, are as their bases are 5 e, and 6. p xij. And 15 Those which are reciprocal in base and height are equal 9 p xij. These consectaries are drawn out of the 16, 18 e. iiij. 16 A tetraedrum is an ordinate pyramid comprehended of four triangles 26. d xj. Therefore 17 The edges of a tetraedrum are six, the plain angles twelve, the solid angles four. For a Tetraedrum is comprehended of four triangles, each of them having three sides, and three corners a piece: And every side is twice taken: Therefore the number of edges is but half so many. And 18 Twelve tetraedra's do fill up a solid place. Because 8. solid right angles filling a place, and 12. angles of the tetraedrum are equal between themselves, seeing that both of them are comprehended of 24. plain rightangles. For a solid right angle is comprehended of three plain right angles: And therefore 8. are comprehended of 24. In like manner the angle of a Tetraedrum is comprehended of three plain equilaters, that is of six third of one right angle: and therefore of two right angles: Therefore 12 are comprehended of 24. And 19 If four ordinate and equal triangles be joined together in solid angles, they shall comprehend a tetraedrum. 20. If a right line whose power is sesquialter unto the side of an equilater triangle, be cut after a double reason, the double segment perpendicular to the centre of the triangle, knit together with the angles thereof shall comprehend a tetraedrum. 13 p xiij. For a solid to be comprehended of right lines understand plains comprehended of right lines, as in other places following. The twenty third Book of Geometry, of a Prisma, 1 A Pyramidate is a plain solid comprehended of pyramids. 2. A pyramidate is a Prisma, or a mingled polyedrum. 3. A prisma is a pyramidate whose opposite plains are equal, alike, and parallel, the rest parallelogramme. 13 dxj Therefore 4. The flattes of a prisma are two more than are the angles in the base. And indeed as the augmentation of a Pyramid from a quaternary is infinite: so is it of a Prisma from a quinary: As if it be from a triangular, quadrangular, or quinquangular base; you shall have a Pentaedrum, Hexaedrum, Heptaedrum, and so in infinite. 5. The plain of the base and height is the solidity of a right prisma. 6. A prisma is the triple of a pyramid of equal base and height. è 7 p. x i i. If the base be triangular, the Prisma may be resolved into prismas of triangular bases, and the theorem shall be concluded as afore. Therefore 7. The plain made of the base and the third part of the height is the solidity of a pyramid of equal base and height. So in the example following, Let 36, the quadrate of 6 the ray, be taken out of 292 9/1156 the quadrate of the side 17 3/34 the side 16 3/34 of 256 9/1156 the remainder shall be the height, whose third part is 5 37/102; the plain of which by the base 72 ¼ shall be 387 11/24 for the solidity of the pyramid given. After this manner you may measure an imperfect Prisma: 8. Homogeneal Prismas of equal height are one to another as their bases are one to another, 29, 30,31, 32 p xj. This element is a consectary out of the 16 e iiij. And 9 If they be reciprocal in base and height, they are equal. This is a Consectary out the 18 e iiij. And 10. If a Prisma be cut by a plain parallel to his opposite flattes, the segments are as the bases are. 25 p. xj. 11. A Prisma is either a Pentaedrum, or Compounded of pentaedra's. Here the resolution showeth the composition. 12 If of two pentaedra's, the one of a triangular base, the other of a parallelogramme base, double unto the triangular, be of equal height, they are equal 40. p xj. The cause is manifest and brief: Because they be the halves of the same prisma: As here thou mayst perccive in a prisma cut into two halves by the diagoni's of the opposite sides. Euclid doth demonstrate it thus: Let the Pentaedra's a e i o u, and y s r l m, be of equal height: the first of a triangular base e i o: The second of a parallelogramme base s l, double unto the triangular. Now let both of them be double and made up, so that first be n● The second y s r l v f. Now again, by the grant, the base s l, is the double of the base e i o,: whose double is th● base e o, by the 12 e x. Therefore the bases s l, and e o, are equal: And therefore seeing the prismas, by the grant, here are of equal height, as the bases by, the conclusion are equal, the prismas are equal; And therefore also their halves a e i o u, and y s n l r, are equal. The measuring of a pentaedrall prisma was even now generally taught: The matter in special may be conceived in these two examples following. The plain of 18. the perimeter of the triangular base, and 12, the height is 216. This added to the triangular base, 15 18/3●. or 15, ⅗, almost twice taken, that is, 31 ⅕, doth make 247 ⅕. for the sum of the whole surface. But the plain of the same base 15 ⅖, and the height 12. is 187 ⅕, for the whole solidity. So in the pentaedrum, the second prisma, which is called Cuneus, (a wedge) of the sharpness, and which also more properly of cutting is called a prisma, the whole surface is 150, and the solidity 90. 13 A prisma compounded of penta●dra's, is either an Hexaedrum or Polyedrum: And the Hexaedrum is either a Parallelepipedum or a Trapezium. 14 A parallelepipedum is that whose opposite plains are parallelogrammes ê 24. p xj. Therefore a Parallelepipedum in solids, answereth to a Parallelogramme in plains. For here the opposite Hedrae or flattes are parallel: There the opposite sides are parallel. Therefore 15 It is cut into two halves with a plain by the diagonies of the opposite sides. 28 p xj. It answereth to the 34. pj. And 16 If it be halfed by two plains halfing the opposite sides, the common bisection and diagony do half one another 39 p xj. 17 If three lines be proportional, the parallelepipedum of mean shall be equal to the equiangled p●rallelepipedum of all them. è 36. p x i. It is a consectary out of the 8 e. 18 Eight rectangled parallelepipeds do fill a solid place. 19 The Figurate of a rectangled parallelepipedum is called a solid, made of three numbers 17. d seven. As if thou shalt multiply 1,2,3. continually, thou shalt make the solid 6. Item if thou shalt in like manner multiply 2,3,4. thou shalt make the solid 24. And the sides of that solid 6 solid shall be 1,2,3. Of 24, they shall be 2,3,4. Therefore 20 If two solids be alike, they have their sides proportionals, and two mean proportionals 21 d seven, 19 21. p viij. It is a consectary out of the 5 e xxij. But the mean proportionals are made of the sides of the like solids, to wit, of the second, third, and fourth: Item of the third, fourth, and fifth, as here tho● seest. Of Geometry the twenty fourth Book. Of a Cube. 1 A Rightangled parallelepipedum is either a Cube, or an Oblong. 2 A Cube is a right angled parallelepipedum of equal flattes, 25. d xj. As here thou seest in these two figures. Therefore 3 The sides of a cube are 12. the plain angles 24. the solid 8. Therefore 4 If six equal quadrates be joined with solid angles, they shall comprehend a cube. As here in these two examples. Therefore 5 If from the angles of a quadrate, perpend●culars equal to the sides be tied together aloft, they shall comprehend a Cube. è 15 p xj. It is a consectary following upon the former consectary: For then shall six equal quadrates be knit together: 6 The diagony of a Cube is of treble power unto the side. For the Diagony of a quadrate is of double power to the side, by the 12 e, xij. And the Diagony of a Cube is of as much power as the side the diagony of the quadrate, by the same e. Therefore it is of treble power to the side. 7 If of four right lines continually proportionally the first be the half of the fourth, the cube of the first shall be the half of the Cube of the second è 33 p xj. It is a consectary out of the 25 e, iiij. From hence Hypocrates first found how to answer Apollo's Problem. 8 The solid plain of a cube is called a Cube, to wit, a solid of equal sides. 19, d seven. Therefore 9 It is made of a number multiplied into his own quadrate. So is a Cube made by multiplying a number by itself, and the product again by the first. Such are these nine first cubes made of the nine first Arithmetical figures. This is the general invention of a Cube, both Geometrical and Arithmetical. 10 If a right line be cut into two segments, the Cube of the whole shall be equal to the Cubes of the segments, and a double solid thrice comprehended of the quadrate of his own segment and the other segment. As for example, the side 12, let it be cut into two segments 10 and 2. The cube of 12. the whole, which is 1728, shall be equal to two cubes 1000, and 8 made of the segments 10. and 2. And a double solid; of which the first 600. is thrice comprehended of 100 the quadrate of his segment 10. and of 2. the other segment: The second 120. is thrice comprehended of 4, the quadrate of his own segment, and of 10. the other segment. Now 1000— 600— 120.— 8, is equal to 1728: And therefore a right etc. Therefore 11. The side of the first several cube is the other side of the second solid: And the quadrate of the same side is the other side of the first solid, whose other side is the side of the second cube; and the quadrate of the same other side is the other side of the second solid. In that equation therefore of four solids with one solid, thou shalt consider a peculiar making and composition: First that the last cube be made of the last segment 2● Then that the second solid of 4, the quadrate of his own segment, and of 10, the other segment be thrice comprehended: Lastly that the first solid of 100, the square of his own segment 10 and the other segment 2, be also thrice comprehended: Lastly, that the Cube 1000, be made of the greater segment 10. Out of this making etc. The plain of the perimeter of the base 20, and the altitude 5 is 100 This added to 25 and 25, both the bases that is to 50, maketh 150, for the whole surface. Now the plain of 25 the base, and the height 5 is 125, for the whole solidity. So in the Oblong, the plain of the base's perimeter 20, and the height 11, is 220, which added to the bases 24 and 24, that is 48, maketh 268, for the whole surface. But the plain of the base 24, and the height 11, is 264, for the solidity. Thus are such kind of walls whether of mud, brick, or stone, of most large houses to be measured. The same manner of Geodesy is also to be used in the measuring of a rhomb, Rhomboides, Trapezium or mensal, and any kind of multangled body. The base is first to be measured, as in the former: Then out of that and the height the solidity shall be manifested: As in the rhomb the base is 24, the height 4. Therefore the solidity is 96. In the Rhomboides, the base is 64 3●/12●: The heigh 11. Therefore the solidity is 1028 44/1●9. The same is the geodesy of a trapezium, as in these examples: The surface of the first is 198: The solidity 192 ½. The surface of the second is 158 3/49; The solidity is 91 29/4●. And from hence also may the capacity or content of vessels or measures, made after any manner of plain solid be esteemed and judged of as here thou seest. For here the plain of the sexangular base is 41 1/7; (For the ray, by the 9 e xviij, is the fide:) and the height 5, shall be 205 5/7. Therefore if a cubical foot do contain 4 quarters, as we commonly call them, then shall the vessel contain 822 6/7 quartes, that is almost 823 quartes. Of Geometry the twenty fifth Book; Of mingled ordinate Polyedra's. 1. A mingled ordinate polyedrum is a pyramidate, compounded of pyramids with their tops meeting in the centre, and their bases only outwardly appearing. SEeing therefore a Mingled ordinate pyramidate is thus made or compounded of pyramids the geodesy of it shall be had from the Geodesy of the pyramids compounding it: And one Base multiplied by the number of all the bases shall make the surface of the body. And one Pyramid by the number of all the pyramids; shall make the solidity. 2 The height of the compounding pyramid is found by the ray of the circle circumscribed about the base, and by the semidiagony of the polyedrum. The base of the pyramid appeareth to the eye: The height lieth hid within, but it is discovered by a right angle triangle, whose base is the semidiagony or half diagony, the shanks the ray of the circle, and the perpendicular of the height. Therefore subtracting the quadrate of the ray, from the quadrate of the halfa diagony the side of the remainder, by the 9 e x ij. shall be the height. But the ray of the circle shall have a special invention, according to the kinds of the base, first of a triangular, and then next of a quinquangular. 3 A mingled ordinate polyedrum hath either a triangular, or a quinquangular base. The division of a Polyhedron ariseth from the bases upon which it standeth. 4 If a quadrate of a triangular base be divided into three parts, the side of the third part shall be the ray of the circle circumscribed about the base. As is manifest by the 12 e. xviij. And this is the invention or way to find out the circular ray for an octoedrum, and an icosoedrum. 5 A mingled ordinate polyedrum of a triangular base, is either an Octoedrum, or an Icosoedrum. This division also ariseth from the bases of the figures. 6 An octoedrum is a mingled ordinate polyedrum, which is comprehended of eight triangles. 27 d xj. As here thou seest, in this Monogrammum and solidum, that is lined and solid octahedrum. Therefore 7 The sides of an octoedrum are 12. the plain angles 24, and the solid 6. And 8 Nine octoedra's do fill a solid place. For four angles of a Tetraedrum are equal to three angles of the Octoedrum: And therefore 12. are equal to nine. Therefore nine angles of an octaedrum do countervail eight solid right angles. And 9 If eight triangles, equilaters and equal be joined together by their edges; they shall comprehend an octaedrum. 10 If a right line on each side perpendicular to the centre of a quadrate and equal to the half diagony be tied together with the angles, it shall comprehend an octaedrum, 14. d xiij. Therefore 11 The Diagony of an octaedrum is of double power to the side. As is manifest by the 9 e xij. And 12 If the quadrate of the side of an octaedrum, be doubled, the side of the double shall be the diagony. As in the figure following, the side is 6. The quadrate 36. the double is 72. whose side 8 8/17. is the diagony. 13 An Icosaedrum is an ordinate polyedrum comprehended of 20 triangles 29 d xj. Therefore 14 The sides of an Icosaedrum are 30. the plain angles 60. the solid 12. 15 If twenty ordinate and equal triangles be joined with solid angles, they shall comprehend an Icosaedrum. 16 If ordinate figures, to wit, a double quinquangle, and one decangle be so inscribed into the same circle, that the side of both the quinquangle do subtend two sides of the decangle, six right lines perpendicular to the circle and equal to his ray, five from the angles of one of the quinquangles, knit together both between themselves, and with the angles of the other quinquangle; the sixth from the centre on each side continued with the side of the decangle, and knit therewith the five perpendiculars, here with the angles of the second quinquangle, they shall comprehend an icosaedrum. è 15 p xiij. In like manner also shall it be proved of the five upper triangles, by drawing the right lines d y, and c n, which as afore (because they knit together equal parallels, to wit, d c, and y n) they shall be equal. But d y, is the side of a sexangle: Therefore c n, shall be also the side of a sexangle: And c g, is the side of a decangle: Therefore a n, whose power is equal to both theirs, by the 9 e xij. shall by the converse of the 15 e xviij, be the side of a quinquangle: And in like manner g t, shall be concluded to be the side of a quinquangle. Wherefore n g t, is an equilater: And the four other shall likewise be equilaters. The other five triangles beneath shall after the like manner be concluded to be equilaters. Therefore one shall be for all, to wit, i b e, by drawing the rays d i, and d e. For i b, whose power, as afore, is as much as the sides of the sexangle, and decangle, shall be the side of the quinquangle: And in like sort b e, being of equal power with d e, and d o, the sides of the sexangle and decangle, shall be the side of the quinquangle. Wherefore the triangle e b i, is an equilater: And the four other in like manner may be showed to be equilaters. Therefore all the side of the twenty triangles, seeing they are equal, they shall be equilater triangles: And by the 8 e, seven. equal. 17 The diagony of an icosaedrum is irrational unto the side. This is the fourth example of irrationality, or incommensurability. The first was of the Diagony and side of a square or quadrate. The second was of the segments of a line proportionally cut. The third of the Diameter of a circle and the side of a quinquangle. And 18 The power of the diagony of an icosaedrum is five times as much as the ray of the circle. 19 A mingled ordinate polyedrum of a quinquangular base is that which is comprehended of 12 quinquangles, and it is called a Dodecaedrum. Therefore 20. The sides of a Dodecaedrum are 30, the plain angles 60. the solid 20. And 21. If 12 ordinate equal quinquangles be joined with solid angles, they shall comprehend a Dodecaedrum. As here thou seest. 22. If the sides of a cube be with right lines cut into two equal parts, and three bisegments of the bisecants in the abbuting plains, neither meeting one the other, nor parallel one unto another, two of one, the third of that next unto the remainder, be so proportionally cut that the lesser segments do bound the bisecant: three lines without the cube perpendicular unto the said plains from the points of the proportional sections, equal to the greater segment knit together, two of the same bisecant, between themselves and with the next angles of cube; the third with the same angles, they shall comprehend a dodecaedrum. 17 p xiij. I say also that it is a Plain quinquangle: For it may be said to be an oblique quinquangle; and to be seated in two plains. Let therefore f h be parallel to d b, and c p: and be equal unto them. And let h z, be drawn: This h z shall be cut one line, by the 14 e seven. For as the whole tr, that is r f, is unto the greater segment that is to f h: so f h, that is z g, is unto g r. And two pair of shanks f h, g r, f c, g z, by the 6 e xxj, are alternely or crossewise parallel. Therefore their bases are continual. Hitherto it hath been proved that the quinquangle made is an equilater and plain: It remaineth that it be proved to be Equiangled. Let therefore the right lines e p, and e c, be drawn: I say that the angles, p b e, and e z i, are equal: Because they have by the construction, the bases of equal shanks equal, being to wit in value the quadruple of l e. For the right line l f, cut proportionally, and increased with the greater segment d f, that is f c, is cut also proportionally, by the 4 e xiv, and by the 7 e xiv, the whole line proportionally cut, and the lesser segment, that is c p, are of treble value to the greater f l, that is of the said l e. Therefore e l, and l c, that is e c, and c p, that is e p, is of quadruple power to e l: And therefore by the 14 e xij, it is the double of it: And e i, itself in like manner, by the fabric or construction, is the double of the same. Therefore the bases are equal. And after the same manner, by drawing the right lines i d, and i b, the third angle b p i, shall be concluded to be equal to the angle e z i. Therefore by the 13 e xiv, five angles are equal. 23. The Diagony is irrational unto the side of the dodecahedrum. This is the fifth example of irrationality and incommensurability. The first was of the diagony and side of a quadrate or square. The second was of a line proportionally cut and his segments: The third is of the diameter of a Circle and the side of an inscribed quinquangle. The fourth was of the diagony and side of an icosahedrum. The fifth now is of the diagony and side of a dodecahedrum. 24 If the side of a cube be cut proportionally, the greater segment shall be the side of a dodecahedrum. The semidiagony and ray of the circle thus found, the altitude remaineth. Take out therefore the quadrate of the ray of the circle, 16 4/225 out of the quadrate of the semidiagony 47. 12458/ 17161. the side of the remainder 3● 2●14406/3861225 is for the altitude or height: whose ⅓ is 5/3. The quinquangled base is almost 38. Which multiplied by 5/3 doth make 63 ⅓ for the solidity of one Pyramid; which multiplied by 12, doth make 760. for the solidity of the whole dodetacedrum. 25 There are but five ordinate solid plains. This appeareth plainly out of the nature of a solid angle, by the kinds of plain figures. Of two plain angles a solid angle cannot be comprehended. Of three angles of an ordinate triangle is the angle of a Tetrahedrum comprehended: Of four, an Octahedrum: Of five, an Icosahedrum: Of six none can be comprehended: For six such like plain angles, are equal to 12 thirds of one right angle, that is to four right angles. But plain angles making a solid angle, are lesser than four right angles, by the 8 e xxij. Of seven therefore, and of more it is, much less possible. Of three quadrate angel's the angle of a cube is comprehended: Of 4. such angel's none may be comprehended for the same cause. Of three angles of an ordinate quinquangle, is made the angle of a Dodecahedrum. Of 4. none may possibly be made: For every such angle: For every one of them severally do countervail one right angle and 1/5 of the same, Therefore they would be four, and three fifths. Of more therefore much less may it be possible. This demonstration doth indeed very accurately and manifestly appear, Although there may be an innumerable sort of ordinate plains, yet of the kinds of angles five only ordinate bodies may be made; From whence the Tetrahedrum, Octahedrum, and Icosahedrum are made upon a triangular base: the Cube upon a quadrangular: And the Dodecahedrum, upon a quinquangular. Of Geometry the twenty sixth Book; Of a Sphere. 1 AN embossed solid is that which is comprehended of an embossed surface. 2. And it is either a sphere or a Mingled form. 3. A sphere is a round imbossement. Therefore 4. A Sphere is made by the conversion of a semicircle, the diameter standing still. 14 d xj. As here thou seest: 5. The greatest circle of a sphere, is that which cutteth the sphere into two equal parts. Therefore 6. That circle which is nearest to the greatest, is greater than that which is farther off. And 7. Those which are equally distant from the greatest are equal. As in the example above written. 8. The plain of the diameter and sixth part of the sphearicall is the solidity of the sphere. Therefore 9 As 21 is unto 11, so is the cube of the diameter unto the sphere. As here the Cube of 14 is 2744. For it was an easy matter for him that will compare the cube 2744, with the sphere, to find that 2744 to be to 1437 ⅓ in the least bounds of the same reason, as 21 is unto 11. Thus much therefore of the Geodesy of the sphere: The geodesy of the Setour and section of the sphere shall follow in the next place. And 10. The plain of the ray, and of the sixth part of the sphearicall is the hemisphere. But it is more accurate and preciser cause to take the half of the sphere. 11. Spheres have a trebled reason of their diameters. So before it was told you; That circles were one to another, as the squares of their diameters were one to another, because they were like plains: And the diameters in circles were, as now they are in spheres, the homologal sides. Therefore seeing that spheres are figures alike, and of treble dimension, they have a trebled reason of their diameters. 12 The five ordinate bodies are inscribed into the same sphere, by the conversion of a semicircle having for the diameter, in a tetrahedrum, a right line of value sesquialter unto the side of the said tetrahedrum; in the other four ordinate bodies, the diagony of the same ordinate. The adscription of ordinate plainebodies is unto a sphere, as before the Adscription plain surfaces was into a circle; of a triangle, I mean, and ordinate triangulate, as Quadrangle, Quinquangle, Sexangle, Decangle, and Quindecangle. But indeed the Geometer hath both inscribed and circumscribed those plain figures within a circle. But these five ordinate bodies, and over and above the Polyhedrum the Stereometer hath only inscribed within the sphere. The Polyhedrum we have passed over, and we purpose only to touch the other ordinate bodies. 13 Out of the reason of the axletree of the sphearicall the sides of the tetraedrum, cube, octahedrum and dodecahedrum are found out. If the same axis be cut into two halves, as in u: And the perpendicular u y, be erected: And y, and a, be knit together, the same y a, thus knitting them, shall be the side of the Octahedrum, as is manifest in like manner, by the said 10 e, viij. and 25 e iiij. The side of the Icosahedrum is had by this means. 14 If a right line equal to the axis of the sphearicall, and to it from the end of the perpendicular be knit unto the centre, a right line drawn from the cutting of the periphery unto the said end shall be the side of the Icosahedrum. 15 Of the five ordinate bodies inscribed into the same sphere, the tetrahedrum in respect of the greatness o● his side is first, the Octahedrum, the second; the Cube, the third; the Icosahedrum, the fourth; and the Dodecahedrum, the fifth. The latter, Euclid doth demonstrate with a greater circumstance. Therefore out of the former figures and demonstrations, let here be repeated, The sections of the axis first into a double reason in s: And the side of the sexangle r l: And the side of the Decangle a r, inscribed into the same circle, circumscribing the quinquangle of an icosahedrum: And the perpendiculars i s, and u l. Here the two triangles a i e, and i e s, are by the 8 e, viij. alike; And as s e, is unto e i: So is i e, unto e a: And by 25 e, iiij, as s e, is to e a: so is the quadrate of s e, to the quadrate of e i: And inversly or backward, as a e, is to s e: so is the quadrate of i e, to the quadrate of s e. But a e, is the triple of s e. Therefore the quadrate of i e, is the triple of s e. But the quadrate of a s, by the grant, and 14 e xij, is the quadruple of the quadrate of s e. Therefore also it is greater than the quadrate of i e: And the right line a s, is greater than i e, and a l, therefore is much greater. But a l, is by the grant compounded of the sides of the sexangle and decangle r l, and a r. Therefore by the 1 c. 5 e, 18. it is cut proportionally: And the greater segment is the side of the sexangle, to wit, r l: And the greater segment of i e, proportionally also cut, is the e. Therefore the said r l, is greeter than y e: And even now it was showed u l, was equal to r l. Therefore u l. is greater than y e: But u e, the side of the Icosahedrum, by 22. e vi. is greater than u l. Therefore the side of the Icosahedrum is much greater, than the side of the dodecahedrum. Of Geometry the twenty seventh Book; Of the Cone and Cylinder. 1 A mingled solid is that which is comprehended of a variable surface and of a base. FOr here the base is to be added to the variable surface. 2 If variable solids have their axes proportional to their bases, they are alike. 24. d xj. It is a Consectary out of the 19 e, iiij. For here the axes and diameters are, as it were, the shanks of equal angles, to wit, of right angles in the base, and perpendicular axis. 3 A mingled body is a Cone or a Cylinder. The cause of this division of a varied or mingled body, is to be conceived from the division of surfaces. 4 A Cone is that which is comprehended of a conical and a base. Therefore 5 It is made by the turning about of a rightangled triangle, the one shank standing still. As it appeareth out of the definition of a variable body. And 6 A Cone is rightangled, if the shank standing still be equal to that turned about: It is Obtusangeld, if it be less: and acutangled, if it be greater. ê 18 d xj. And 7 A Cone is the first of all variable. For a Cone is so the first in variable solids, as a triangle is in rectilineall plains: As a Pyramid is in solid plains; For neither may it indeed be divided into any other variable solids more simple. And 8 Cones of equal height are as their bases are 11. p xij. As here you see. And 9 They which are reciprocal in base and height are equal, 15 p x ij. These are consectaries drawn out of the 12 and 13 e, iiij. As here you see. 10 A Cylinder is that which is comprehended of a cyliudricall surface and the opposite bases. Therefore 11 It is made by the turning about of a right angled parallelogramme, the one side standing still. 21. dxj As is apparent out the same definition of a varium. 12. A plain made of the base and height is the solidity of a Cylinder. This manner of measuring doth answer, I say, to the manner of measuring of a prisma, and in all respects to the geodesy of a right angled parallelogramme. If the cylinder in the opposite bases be oblique, then if what thou cuttest off from one base thou dost add unto the other, thou shalt have the measure of the whole, as here thou seest in these cylinders, a and b. As here the diameter of the inner Circle is 6 foot: The periphery is 18 6/7: Therefore the plot or content of the circle is 28 2/7 Of which, and the height 10, the plain is 282 6/7 for the capacity of the vessel. Thus therefore shalt thou judge, as afore, how much liquor or any thing esle contained, a cubical foot may hold. 13. A Cylinder is the triple of a cone equal to it in base and height. 10 p xij. The demonstration of this proposition hath much troubled the interpreters. The reason of a Cylinder unto a Cone, may more easily be assumed from the reason of a Prism unto a Pyramid: For a Cylinder doth as much resemble a Prism, as the Cone doth a Pyramid: Yea and within the same sides may a Prism and a Cylinder, a Pyramid and a Cone be contained: And if a Prism and a Pyramid have a very multangled base, the Prism and Clinder, as also the Pyramid and Cone, do seem to be the same figure. Lastly within the same sides, as the Cones and Cylinders, so the Prisma and Pyramids, from their axeletrees and diameters may have the similitude of their bases. And with as great reason may the Geometer demand to have it granted him, That the Cylinder is the treble of a Cone● As it was demanded and granted him, That Cylinders and Cones are alike, whose axletees are proportional to the diameters of their bases. Therefore 14. A plain made of the base and third part of the height, is the solidity of the cone of equal base & height, Of two cones of one common base is made Archimede's Rhombus, as here, whose geodaesy shall be cut of two cones. And 15. Cylinder of equal height are as their bases are. 11 p xij. And 16 Cylinders reciprocal in base and height are eequall. 15 p xij. Both these affections are in common attributed to the equally manifold of first figures. And 17. If a Cylinder be cut with a plain surface parallel to his opposite bases, the segments are, as their axes are 13 p xij. The unequal sections of a sphere we have reserved for this place: Because they are comprehended of a surface both sphearicall and conical, as is the sectour. As also of a plain and sphearicall, as is the section: And in both like as in a Circle, there is but a greater and lesser segment. And the sectour, as before, is considered in the centre. 18. The sectour of a sphere is a segment of a sphere, which without is comprehended of a sphearicall● within of a conical bounded in the centre, the greater of a concave, the lesser of a convex. Archimedes', maketh mention of such kind of Sectours, in his 1 book of the Sphere. From hence also is the geodesy following drawn. And here also is there a certain analogy with a circular sectour. 19 A plain made of the diameter, and sixth part of the greater, or lesser sphearicall, is the greater or lesser sector. And from hence lastly doth arise the solidity of the section, by addition and subduction. 20. If the greater sectour be increased with the internal cone, the whole shall be the greater section: If the lesser be diminished by it, the remain shall be the lesser section. As here the inner cone measured is 126 4/63. The greater sectour, by the former was 1026 ⅔. And 126 ⅔— 126 4/63 do make 1152 46/63. Again the lesser sectour, by the next precedent, was 410 ⅔: And here the inner cone is 126 4/63 And therefore 410 2/●— 126 4/63 that is 284 38/63 is the lesser section. FINIS.