The pathway to KNOWLEDGE, CONTAINING THE FIRST PRINciples of Geometry, as they may most aptly be applied unto practice, both for use of instruments Geometrical, and astronomical and also for projection of plats in every kind, and therefore much necessary for all sorts of men.  

Geometries verdict  
All fresh fine wits by me are filled,  
All gross dull wits wish me exiled:  
Though no man's wit reject will I,  
Yet as they be, I will them try.    

The arguments of the four books  
The first book declareth the definitions of the terms and names used in Geometry, with certain of the chief grounds whereon the art is founded. And then teacheth those conclusions, which may serve diversely in all works Geometrical.  
The second book doth set forth the Theorems, (which may be called approved truths) serving for the due knowledge and sure proof of all conclusions and works in geometry.  
The third book entreateth of divers forms, and sundry protractions thereto belonging, with the use of certain conclusions.  
The fourth book teacheth the right order of measuring all plat forms, and bodies also, by reason Geometrical.   

TO THE GENTLE READER.  
EXCUSE ME, GENTLE REder if ought be amiss, strange paths are not trodden all truly at the first: the way must needs be cumbrous, where none hath gone before. where no man hath given light, light is it to offend, but when the light is showed ones, light is it to amend. If my light may so light some other, to espy and mark my faults, I wish it may so lighten them, that they may void offence. Of staggering and stomblinge, and unpleasant turmoiling: often offending, and seldom amending, such vices to eschew, and their fine wits to show that they may win the praise, and I to hold the candle, whilst they their glorious works with eloquence set forth, so cunningly invented, so finely indicted, that my books may seem worthy to occupy no room. For neither is mi wit so finely filled, neither mi learning so largely lettered, neither yet mi laiser so quiet and uncombered, that I may perform justly so learned a labour or accordingly to accomplish so halt an enforcement, yet may I think thus: This candle did I light: this light have I kindled: that learned men may see, to practise their pens, their eloquence to advance, to register their names in the book of memory I drew the plat rudely, whereon they may build, whom god hath endued with learning and livelihood. For living by labour doth learning so hinder, that learning serveth living, which is a perverse trade. Yet as careful family shall cease her cruel calling, and suffer any laiser to learning to repair, I will not cease from travail the path so to trade, that finer wits may fashion themselves with such glimpsing dull light, a more complete work at laiser to finish, with invention agreeable, and aptness of eloquence.  
And this gentle reader I heartily protest where error hath happened I wish it redressed.   

TO THE MOST NOble and puissant prince Edward the sixth by the grace of God, of England France and Ireland king, defender of the faith, and of the Church of England and Ireland in earth the supreme head.  
IT IS NOT Unknown to your majesty, most sovereign lord, what great disceptation hath been amongst the witty men of all nations, for the exact knowledge of true felicity, both what it is, and wherein it consisteth: touching which thing, their opinions almost were as many in numbered, as were the persons of them, that either disputed or wrote thereof. But and if the diversity of opinions in the vulgar sort for placing of their felicity shall be considered also, the variety shall be found so great, and the opinions so dissonant, yea plainly monstrous, that no honest wit would vouchsafe to lose time in hearing them, or rather (as I may say) no wit is of so exact remembrance, that can consider together the monstrous multitude of them all. And yet not withstanding this repugnant diversity, in two things do they all agree. First all do agree, that felicity is and aught to be the stop and end of all their doings, so that he that hath a full desire to any thing, how so ever it be esteemed of other men, yet he esteemeth himself happy, if he may obtain it: and contrary ways unhappy if he can not attain it. And therefore do all men put their whole study to get that thing, wherein they have persuaded themself that felicity, doth consist. Wherefore some which put their felicity in feeding their bellies, think no pain to be hard, nor no deed to be unhonest, that may be a means to fill that foul pauche. Other which put their felicity in play and idle pastimes, judge no time evil spent, that is employed thereabout: nor no fraud unlawful that may further their winning. If I should particularly overrun but the common sorts of men, which put their felicity in their desires, it would make a great book of itself. Therefore will I let them all go, and conclude as I began, That all men employ their whole endeavour to that thing, wherein they think felicity to stand. which thing who so listeth to mark exactly, shall be able to espy and judge the natures of all men, whose conversation he doth know, though they use great dissimulation to colour their desires, especially when they perceive other men to mislike that which they so much desire: For no man would gladly have his appetite improved. And hereof cometh that seconnde. thing wherein all agree, that every man would most gladly win all other men to his sect, and to make them of his opinion, and as far as he dare, will dispraise all other men's judgements, and praise his own ways only, unless it be when he dissimuleth, and that for the furtherance of his own purpose. And this property also doth give great light to the full knowledge of men's natures, which as all men ought to observe, so princes above other have most cause to mark for sundry occasions which may lie them on, whereof I shall not need to speak any farther, considering not only the greatness of wit, and exactness of judgement which god hath lent unto your highness person, but also the most grave wisdom and profound knowledge of your majesties most honourable council, by whom your highness may so sufficiently understand all things convenient, that less shall it need to understand by private reading, but yet not utterly to refuse to read as often as occasion may serve, for books dare speak, when men fear to displease. But to return again to my first matter, if none other good thing may be learned at their manners, which so wrongfully place their felicity, in so miserable a condition (that while they think themselves happy, their felicity must needs seem unlucky, to be by them so evil placed) yet this may men learn at them, by those two spectacles to espy the secret natures and dispositions of others, which thing unto a wise man is much available. And thus will I omit this great tablement of unhappy hap, and will come to three other sorts of a better degree, whereof the one putteth felicity to consist in power and royalty. The second sort unto power annexeth worldly wisdom, thinking him full happy, that could attain those two, whereby he might not only have knowledge in all things, but also power to bring his desires to end. The third sort esteemeth true felicity to consist in wisdom annexed with virtuous manners, thinking that they can take harm of nothing, if they can with their rwysedome overcome all vices. Of the first of those th●ee sorts there hath been a great numbered in all ages, yea many mighty kings and great governors, which cared not greatly how they might achieve their purpose, so that they did prevail: nor did not take any greater care for governance, then to keep the people in only fear of them, Whose common sentence was always this: Oderint dum metuant. And what good success such men had, all histories do report. Yet have they not wanted excuses: yea julius Caesar (which in deed was of the second sort) maketh a kind of excuse by his common sentence, for them of that first sort, for he was ever wont to say: 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉. Which sentence I wish had never been learned out of Grecia. But now to speak of the second sort, of which there hath been very many also, yet for this present time amongst them all, I will take the examples of king Phylippe of Macedon, and of Alexander his son, that valiant conqueror. First of king philip it appeareth by his letter sent unto Aristotle that famous philosopher, that he more delighted in the birth of his son, for the hope of learning and good education, that might happen to him by the said Aristotle, than he did rejoice in the continuance of his succession, for these were his words and his whole epistle, worthy to be remembered and registered every where.  
〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉.  
〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉.  
That is thus in sense,  
Philip unto Aristotle sendeth greeting.  
You shall understand, that I have a son borne, for which cause I yield unto God most hearty thanks, not so much for the birth of the child, as that it was his chance to be borne in your time. For my trust is, that he shall be so brought up and instructed by you, that he shall become worthy not only to be named our son, but also to be the successor of our affairs.  
And his good desire was not all vain, for it appeared that Alexander was never so busied with wars (yet was he never out of most terrible battle) but that in the mids thereof he had in remembrance his studies, and caused in all countries as he went, all strange beasts, fowls and fishes, to be taken and kept for the aid of that knowledge, which he learned of Aristotle: And also be had with him always a great numbered of learned men. And in the most busy time of all his wars against Darius' king of Persia, when he hard that Aristotle had put forth certain kookes of such knowledge wherein he had before studied, he was offended with Aristotle, and wrote to him this letter. 
〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉.  
〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉.  that is Alexander unto Aristotle sendeth greeting.  
You have not done well, to put forth those books of secret philosophy entitled, 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 For wherein shall we excel other, if that knowledge that we have studied, shall be made comen to all other men, namely sith our desire is to excel other men in experience and knowledge, rather than in power and strength. Farewell.  
By which letter it appeareth that he esteemed learning and knowledge above power of men. And the like judgement did he utter, when he beheld the state of Diogenes Cinicus, adiudginge it the best state next to his own, so that he said: If I were not Alexander, I would wish to be Diogenes. Whereby appeareth, how he esteemed learning, and what felicity he put therein, reputing all the world save himself to be inferior to Diogenes. And by all conjectures, Alexander did esteem Diogenes one of them which contemned the vain estimation of the deceitful world, and put his whole felicity in knowledge of virtue, and practise of the same, though some report that he knew more virtue than he followed: But whatsoever he was, it appeareth that Socrates and Plato and many other did forsake their livings and sell away their patrimony, to the intent to seek and travail for learning, which examples I shall not need to repeat to your Majesty, partly for that your highness doth often read them and other like, and partly sith your majesty hath at hand such learned schoolmasters, which can much better than I, declare them unto your highness, and that more largely also then the shortness of this epistle will permit. But this may I yet add, that King Solomon whose renown spread so far abroad, was very greatly esteemed for his wonderful power and exceeding treasure, but yet much more was he esteemed for his wisdom And himself doth bear witness, that wisdom is better than precious stones. yea all things that can be desired are not to be compared to it. But what needeth to allege one sentence of him, whose books altogether do none other thing, than set forth the praise of wisdom & knowledge? And his father king David joineth virtuous conversation and knowledge together, as the sum of perfection and chief felicity. Wherefore I may justly conclude, that true felicity doth consist in wisdom and virtue. Then if wisdom be as Cicero defineth it, Divinarum atque humanarum rerum scientia, then ought all men to travail for knowledge in matters both of religion and humane doctrine, if he shall be counted wise, and able to attain true felicity: But as the study of religious matters is most principal, so I leave it for this time to them that better can write of it then I can. And for humane knowledge this will I boldly say, that who soever will attain true judgement therein, must not only travail in the knowledge of the tongues, but must also before all other arts, taste of the mathematical sciences, specially Arithmetic and Geometry, without which it is not possible to attain full knowledge in any art. Which may sufficiently by gathered by Aristotle notonly in his books of demonstration (which can not be understand without Geometry) but also in all his other works. And before him Plato his master wrote this sentence on his school house door. 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉. Let no man entre here (saith he) without knowledge in Geometry. Wherefore most mighty prince, as your most excellent Majesty appeareth to be borne unto most perfect felicity, not only by reason that God moved with the long prayers of this realm, did send your highness as a most comfortable inheritor to the same, but also in that your Majesty was borne in the time of such skilful schoolmasters & learned teachers, as your highness doth not a little rejoice in, and profit by them in all kind of virtue & knowledge. Among which is that heavenvly knowledge most worthily to be praised, wherbi the blindness of error & superstition is exiled, & good hope conceived that all the sedes & fruits thereof, with all kinds of vice & iniquity, whereby virtue is hindered, & justice defaced, shall be clean extrirped and rooted out of this realm, which hope shall increase more and more, if it may appear that learning be esteemed & flourish within this realm. And all be it the chief learnig be the divine scriptures, which in struck the mind principally, & next thereto the laws politic, which most specially defend the right of goods, yet is it not possible that those two can long be well used, if that aid want that governeth health and expelleth sickness, which thing is done by physic, & these require the help of the seven. liberal sciences, but of none more than of arithmetic and Geometry, by which not only great things are wrought touching accounts in all kinds, & in surveying & measuring of lands, but also all arts depend partly of them, & building which is most necessary can not be without them, which thing considering, moved me to help to serve your majesty in this point as well as other ways, & to do what may be in me, that not only they which studi princicpally for lernig, may have furderance by mi poor help, but also those which have no time to travail for exacter knowledge, may have some help to understand in those Mathematical arts, in which as I have all ready set forth somewhat of Arithmetic, so god willing I in tend shortly to set forth a more exacter work thereof. And in the mean season for a taste of Geometry, I have set forth this small introduction, desiring your grace not so much to behold the simpleness of the work, in comparison to your majesties excellency,, as to favour the edition thereof, for the aid of your humble subjects, which shall think themselves more and more daily bounden to your highness, if when they shall perceive your graces desire to have them profited in all knowledge and virtue. And I for my poor ability considering your majesties study for the increase of learning generally through all your highness' dominions, and namely in the universities of Orforde and Camebridge, as I have an earnest good will as far as my simple service and small knowledge will suffice, to help toward the satisfying of your grace's desire, so if I shall perceive that my service may be to your majesties contentation, I will not only put forth the other two books, which should have been set forth with these two, if misfortune had not hindered it, but also I will set forth other books of more exacter art, both in the Latin tongue and also in the english, whereof part be all ready written, and new instruments to them devised, and the residue shall be ended with all possible speed. I was boldened to dedicate this book of geometry unto your Majesty, not so much because it is the first that ever was set forth in English, and therefore for the novelty a strange present, but for that I was persuaded, that such a wise prince doth desire to have a wise sort of subjects. For it is a kings chief rejoicing and glory; if his subjects be rich in substance, and witty in knowledge: and contrary ways nothing can be more grievous to a noble king, then that his realm should be other beggarly or full of ignorance: But as god hath given your grace a realm both rich in commodities and also full of witty men, so I trust by the reading of witty arts (which be as the whet stones of wit) they must needs increase more and more in wisdom, and peradventure find some thing toward the aid of their substance, whereby your grace shall have new occasion to rejoice, seeing your subjects to increase in substance or wisdom, or in both. And they again shall have new and new causes to pray for your majesty, perceiving so gracious a mind toward their benefit. And I trust (as I desire) that a great numbered of gentlemen, especially about the court, which understand not the latin tongue, or else for the hardness of the matter could not away with other men's writing, will fall in trade with this easy form of teaching in their vulgar tongue, and so employ some of their time in honest study, which were wont to bestow most part of their time in trifling pastime: For undoubtedly if the mean other your majesties service, other their own wisdom, they will be content to employ some time about this honest and witty exercise. For whose encouragement to the intent they may perceive what shall be the use of this science, I have not only written somewhat of the use of Geometry, but also I have annexed to this book the names and brief arguments of those other books which I will set forth hereafter, and that as shortly as it shall appear unto your majesty by conjecture of their diligent using of this first book, that they will use well the other books also. In the mean season, and at all times I will be a continual petitioner, that god may work in all english hearts an earnest mind to all honest exercises, whereby they may serve the better your majesty and the realm. And for your highness I beseech the most merciful god, as he hath most favourably sent you unto us, as our chief comforter in earth, so that he will increase your majesty daily in all virtue and honour with most prosperous● success, and augment in us your most humble subjects, true love to godward, and just obedience to ward your highness with all reverence and subjection.  
At London the xxviij day of januarie. M. D. L I Your majesties most humble servant and obedient subject, Robert Record.   

THE PREFACE, declaring briefly the commodities of geometry, and the necessity thereof.  
geometry may think itself to sustain great injury, if it shall be enforced other to show her manifold commodities, or else not to press into the sight of men, and therefore might this ways answer briefly: Other I am able to do you much good, or else but little. If I be able to do you much good, then be you not your own friends, but greatly your own enemies to make so little of me, which may profit you so much. For if I were as uncourteous as you unkind, I should utterly refuse to do them any good, which will so curiously put me to the trial and proof of my commodities, or else to suffer exile, and namely sith I shall only yield benefits to other, and receive none again. But and if you could say truly, that my benefits be neither many nor yet great, yet if they be any, I do yield more to you, than I do receive again of you, and therefore I ought not to be repelled of them that love themself, although they love me not at all for myself. But as I am in nature a liberal science, so can I not against nature contend with your inhumanity, but must show myself liberal even to mine enemies. Yet this is my comfort again, that I have none enemies but them that know me not, and therefore may hurt themselves, but can not noye me. If they dispraise the thing that they know not, all wise men will blame them and not credit them. and if they think they know me, let them show one untruth and error in me, and I will give the victory.  
Yet can no human science say thus, but I only, that there is no spark of untruth in me: but all my doctrine and works are without any blemish of error that man's reason can discern. And next unto me in certainty are my three sisters, Arithmetic, Music, and Astronomy, which are also so near knit in amity, that he that loveth the one, can not despise the other, and in especial Geometry, of which not only these three, but all other arts do borrow great aid, as partly hereafter shall be showed. But first will I begin with the unlearned sort, that you may perceive how that no art can stand without me. For if I should declare how many ways my help is used, in measuring of ground, for meadow, corn, and wood: in hedging, in diching, and in stacks making, I think the poor Husband man would be more thankful unto me, than he is now, whyses he thinketh that he hath small benefit by me. Yet this may he conjecture certainly, that if he keep not the rules of Geometry, he can not measure any ground truly. And in diching, if he keep not a proportion of breadth in the mouth, to the breadth of the bottom, and just slopeness in the sides agreeable to them both, the ditch shall be faulty many ways. When he doth make stacks for corn, or for hay, he practiseth good Geometry, else would they not long stand: So that in some stacks, which stand on four pillars, and yet made round, do increase greater and greater a good height, and then again turn smaller and smaller unto the top: you may see so good Geometry, that it were very difficult to counterfeit the like in any kind of building. As for other infinite ways that he useth my benefit, I overpass for shortness.  
Carpenters, carvers, joiners, and Masons, do willingly acknowledge that they can work nothing without reason of Geometry, in so much that they challenge me as a peculiar science for them. But in that they should do wrong to all other men, seeing every kind of men have some benefit by me, not only in building, which is but other men's costs, and the art of Carpenters, Masons, and the other aforesaid, but in their own private profession, whereof to avoid tediousness I make this rehearsal.  

Sith merchants by ships great riches do win,  
I may with good right at their feat begin.  
The Ships on the sea with Sail and with Ore,  
were first found, and still made, by Geometries lore,  
Their Compass, their Card, their Pull is, their Ankers,  
were found by the skill of witty Geometers.  
To set forth the Capstocke, and each other part,  
would make a great show of Geometries art.  
Carpenters, Carvers, joiners and Masons,  
Painters and Limners with such occupations,  
Broderers, Goldsmiths, if they be cunning,  
Must yield to Geometrye thanks for their learning.  
The Cart and the Plough, who doth them well mark,  
Are made by good geometry. And so in the work  
Of Tailors and Shoemakers, in all shapes and fashion,  
The work is not praised, if it want proportion.  
So we avers by geometry had their foundation,  
Their Loom is a frame of strange imagination.  
The wheel that doth spin, the stone that doth grind,  
The Mill that is driven by water or wind,  
Are works of geometry strange in their trade,  
Few could them devise, if they were unmade.  
And all that is wrought by weight or by measure,  
without proof of Geometry can never be sure.  
Clocks that be made the times to divide,  
The wittiest invention that ever was spied,  
Now that they are common they are not regarded,  
The art's man contemned, the work unrewarded.  
But if they were scarce, and one for a show,  
Made by geometry, then should men know,  
That never was art so wonderful witty,  
So needful to man, as is good Geometry.  
The first finding out of every good art,  
Seemed then unto men so godly a part,  
That no recompense might satisfy the finder,  
But to make him a god, and honour him for ever.  
So Ceres and Passas, and Mercury also,  
Eosus and Neptune, and many other more,  
were honoured as gods, because they did teach,  
first tillage and weaving and eloquent speech,  
Or winds to observe, the seas to sail over,  
They were called gods for their good endeavour.  
Then were men more thankful in that golden age:  
This iron world no we ungrateful in rage,  
will yield the thy reward for travail and pain,  
with slanderous reproach, and spiteful disdain.  
Yet though other men unthankful will be,  
Suruayers have cause to make much of me.  
And so have all Lords, that lands do possess:  
But Tennauntes I fear will like me the less.  
Yet do I not wrong but measure all truly,  
And yield the full right to every man justly.  
Proportion Geometrical hath no man oppressed,  
If any be wronged, I wish it redressed.   
But now to proceed with learned professions, in Logic and Rhetoric and all parts of philosophy, there needeth none other proof than Aristotle his testimony, which without Geometry proveth almost nothing. In Logic all his good syllogisms and demonstrations, he declareth by the principles of geometry. In philosophy, neither motion, nor time, nor airy impressions could he aptly declare, but by the help of geometry as his works do witness. Yea the faculties of the mind do the he express by similitude to sigures of geometry. And in moral philosophy he thought that justice could not well be taught, nor yet well executed without proportion geometrical. And this estimation of Geometry he may seem to have learned of his master Plato, which without geometry would teach nothing, neither would admit any to hear him, except he were expert in Geometry. And what marvel if he so much esteemed geometry, seeing his opinion was, that god was alwaaies working king by Geometry? Which sentence plutarch declareth at large. And although Plat o do use the help of geometry in all the most weight matter of a common wealth, yet it is so general in use, that no small things almost can be well done without it. And therefore saith he: that geometry is to be learned, if it were for none other cause, but that all other arts are both sooner and more surely understand by help of it.  
What great help it doth in physic, Galene doth so often and so copiousely declare, that no man which hath red any book almost of his, can be ignorant thereof. in so much that he could never cure well a round ulcere, till reason geometrical did teach it him. Hypocrates is earnest in admonishing that study of geometry must prepare the way to physic, as well as to all other arts.  
I should seem somewhat to tedious, if I should reckon up, how the divines also in all their mysteries of scripture do use help of geometry: and also that lawyers can never understand the hole law, no nor yet the first title there of exactly without Geometry. For if laws can not well be established, nor justice duly executed without geometrical proportion, as both Plato in his Politic books, and Aristotle in his morals do largely declare. Yea sith Lycurgus that chief lawmaker amongst the Lacedæmonians, is most praised for that he did change the state of their common wealth from the proportion Arithmetical to a proportion geometrical, which without knowledge of both he could not do, than is it easy to perceive how necessary Geometry is for the law and students thereof. And if I shall say precisely and freely as I think, he is utterly destitute of all ability to judge in any art, that is not somewhat expert in the Theorems of Geometry.  
And that caused Galene to say of himself, that he could never perceive what a demonstration was, no not so much, as whether there were any or none, till he had by geometry gotten ability to understand it, although he heard the best teachers that were in his time. It should be to long and needless also to declare what help all other arts Mathematical have by geometry, sith it is the ground of all their certainty, and no man studious in them is so doubtful thereof, that he shall need any persuasion to procure credit thereto. For he can not read two lines almost in any mathematical science, but he shall espy the nedefulnes of geometry. But to avoid tediousness I will make an end hereof with that famous sentence of ancient Pythagoras, That who so will travail by learning to attain wyfedome, shall, never approach to any excellency without the arts mathematical, and especially Arithmetic and Geometry.  
And if I shall somewhat speak of noble men, and governors of realms, how needful geometry may be unto them, then must I repeat all that I have said before, sith in them ought all knowledge to abound, namely that may appertain either to good governance in time of peace, either witty policies in time of war. For ministration of good laws in time of peace Lycurgus' example with the testimonies of Plato and Aristotle may suffice. And as for wars, I might think it sufficient that Vegetius hath written, and after him Valturius in commendation of Geometry, for use of wars, but all their words seem to say nothing, in comparison to the example of Archimedes worthy works made by geometry, for the defence of his country, to read the wonderful praise of his witty devices, set forth by the most famous histories of Livius, plutarch, and Pliny, and all other hystoriographiers, which write of the strong siege of Syracuse made by that valiant captain, and noble warrior Marcellus, whose power was so great, that all men marveled how that one city could withstand his wonderful force so long. But much more would they marvel, if they understood that one man only did withstand all Marcellus strength, and with counter engines destroyed his engines to the utter astonishment of Marcellus, and all that were with him. He had invented such balastelas that did shoot out a hundred darts at one shot, to the great destruction of Marcellus soldiers, whereby a fond tale was spread abroad, how that in Syracuse there was a wonderful giant, which had a hundred hands, and could shoot a hundred darts at ones. And as this fable was spread of Archimedes, so many other have been feigned to be giants and monsters, because they did such things, which far passed the wit of the common people. So did they feign Argus to have a hundred eyes, because they heard of his wonderful circumspection, and thought that as it was above their capacity, so it could not be, unless he had a hundred eyes. So imagined they Janus to have two faces, one looking forward, and an other backward, because he could so wittily compare things passed with things that were to come, and so duly pondre them, as if they were all present. Of like reason did they fain Lynceus to have such sharp sight, that he could see through walls and hills, because peradventure he did by natural judgement declare what commodities might be digged out of the ground. And an infinite noumbre like fables are there, which sprang all of like reason.  
For what other thing meaneth the fable of the great giant Atlas, which was imagined to bear up heaven on his shoulders? but that he was a man of so high a wit, that it reached unto the sky, and was so skilful in Astronomy, and could tell before hand of Eclipses, and other like things as truly as though he did rule the stars, and govern the planets.  
So was Aeolus accounted god of the winds, and to have them all in a cave at his pleasure, by reason that he was a witty man in natural knowledge, and observed well the change of wethers, and was the first that taught the observation of the winds. And like reason is to be given of all the old fables.  
But to return again to Archimedes, he did also by art perspective (which is a part of geometry) devise such glasses within the town of Syracuse, that did bourn their enemies ships a great way from the town, which was a marvelous politic thing. And if I should repeat the varietees of such strange inventions, as Archimedes and others have wrought by geometry, I should not only exceed the order of a preface, but I should also speak of such things as can not well be understand in talk, without some knowledge in the principles of geometry.  
But this will I promise, that if I may perceive my pains to be thankfully taken, I will not only write of such pleasant inventions, declaring what they were, but also will teach how a great numbered of them were wrought, that they may be practised in this time also. Whereby shall be plainly perceived, that many things seem impossible to be done, which by art may very well be wrought. And when they be wrought, and the reason thereof not understand, than say the vulgar people, that those things are done by negromancy. And hereof came it that friar Bacon was accounted so great a negromancier, which never used that art (by any conjecture that I can find) but was in geometry and other mathematical sciences so expert, that he could do by them such things as were wonderful in the sight of most people.  
Great talk there is of a glass that he made in Oxford, in which men might see things that were done in other places, and that was judged to be done by power of evil spirits. But I know the reason of it to be good and natural, and to be wrought by geometry (sith perspective is a part of it) and to stand as well with reason as to see your face in common glass. But this conclusion and other divers of like sort, are more meet for princes, for sundry causes, than for other men, and ought not to be taught commonly. Yet to repeat it, I thought good for this cause, that the worthiness of geometry might the better be known, & partly understanding given, what wonderful things may be wrought by it, and so consequently how pleasant it is, and how necessary also.  
And thus for this time I make an end. The reason of some things done in this book, or omitted in the same, you shall find in the preface before the Theorems.   

The definitions of the principles of GEOMETRY.  
GEOMETRY TEAcheth the drawing, Measuring and proportion of figures. but in as much as no figure can be drawn, but it must have certain bounds and enclosures of lines: and every line also is begun and ended at some certain prick, first it shall be meet to know these smaller parts of every figure, that thereby the whole figures may the better be judged, and distinct in sunder.  
A Point or a prick, is named of Geometricians that small and unsensible shape, which hath in it no parts, 
Apoincte.  that is to say: neither length, breadth nor depth. But as this exactness of definition is more meeter for only theoric speculation, then for practise and outward work (considering that mine intent is to apply all these whole principles to work) I think meeter for this purpose, to call a point or prick, that small print of pen, pencyle, or other instrument, which is not moved, nor drawn from his first touch, and therefore hath no notable length nor breadth: as this example doth declare.  
Where I have set three pricks, each of them having both length and breadth, though it be but small, and therefore not notable  
Now of a great numbered of these pricks, is made a Line, as you may perceive by this form ensuing ........................ where as I have set a numbered of pricks, so if you with your pen will set in more other pricks between every two of these, then will it be alyne, as here you may see— and this line, is called of Geometricians, 
Alyne.  Length without breadth.  
But as they in their theorikes (which are only mind works) do precisely understand these definitions, so it shall be sufficient for those men, which seek the use of the same things, as sense may duly judge them, and apply to handy works if they understand them so to be true, that outward sense can find none error therein.  
Of lines there be two principal kinds. The one is called a right or strait line, and the other a crooked line.  
A Strait line, 
A straight line.  is the shortest that may be drawenne between two pricks.  
And all other lines, 
A crooked line.  that go not right forth from prick to prick, but boweth any way such are called Crooked lines as in these examples following ye may see, where I have set but one form of a stratght line, for more forms there be not, but of crooked lines there be innumerable diversities, whereof for example's sum I have set here.  
— A right line.   diagram 
Crooked   diagram 
lines.   diagram 
Crooked lines.   diagram  diagram 
So that when so ever any such meeting of lines doth happen, the place of their meeting is called an Angle or corner.  
Of angles there be three general kinds: a sharp angle, a square angle, and a blunt angle. The square angle, which is commonly named a right corner, 
A right angle.  is made of two lines meeting together in form of a squire, which two lines, if they be drawn forth in length, will cross one an other: as in the examples following you may see.  

A sharp corner.  A sharp angle is so called, because it is lesser than is a square angle, and the lines that make it, do not open so wide in their departing as in a square corner, and if they be drawn cross, all four corners will not be equal.  

Ab lante angle.  A blunt or broad corner, is greater than is a square angle, and his lines do part more in sunder then in a right angle, of which all take these examples.   diagram 
Right angles.   diagram 
Sharp angles.   diagram 
Blunt or broad angles.  
But now as of many pricks there is made one line, so of diverse lines are there made sundry forms, figures, and shapes, which all yet be called by one proper name, plat forms, 
A plat form.  and they have both length and breadth, but yet no deepness.  
And the bounds of every plat form are lines: as by the examples you may perceive.  
Of plat forms some be plain, and some be crooked, and some partly plain, and partly crooked.  

A plain plat.  A plain plat is that, which is made all equal in height, so that the middle parts neither bulk up, neither shrink down more than the both ends.  

A crooked plat.  For when the one part is higher than the other, then is it named a Crooked plat.  
And if it be partly plain, and partly crooked, then is it called a Mixed plat, of all which, these are examples.   diagram 
A plain plat.   diagram 
A crooked plat.   diagram 
A mixed plat.  
Now all bodies have plat forms for their bounds, so in a die (which is called a cubike body) by geomdtricians, 
Lubike. Asheler.  and an ashler of masons, there are vi sides, which are vi plat forms, and are the bounds of the die.  
But in a Globe, (which is a body round as a bowl) there is but one plat form, and one bound, 
A globe.  and these are the examples of them both.   diagram 
A die or ashler.   diagram 
A globe.  
But because you shall not muse what I do call a bound, 
A bound,  I mean thereby a general name, betokening the beginning, end and side, of any form.  

Form, figure.  A form, figure, or sharp, is that thing that is enclosed within one bond or many bonds, so that you understand that shape, that the eye doth discern, and not the substance of the body.  
Of figures there be many sorts, for either they be made of pricks, lines, or plat forms. notwithstanding to speak properly, a figure is ever made by plat forms, and not of bare lines unclosed, neither yet of pricks.  
Yet for the lighter form of teaching, it shall not be unseemly to call all such shapes, forms and figures, which the eye may discern distinctly.  
And first to begin with pricks, there may be made diverse forms of them, as partly here doth follow.   diagram 
Alynearie numbered.   diagram 
Trianguler numbers   diagram 
Long square numbered.   diagram 
Just square numbers   diagram 
a threcornered spire.   diagram 
A square spire.  
And so may there be infinite forms more, which I omit for this time, considering that their knowledge appertaineth more to Arithmetic figural, than to Geometry.  
But yet one name of a prick, which he taketh rather of his place then of his form, may I not overpass. And that is, when a prick standeth in the middle of a circle (as no circle can be made by compass without it) then is it called a centre. And thereof do masons, and other work men call that patron, 
A centre  a centre, whereby they draw the lines, for just hewing of stones for arches, vaults, and chimneys, because the chief use of that patron is wrought by finding that prick or centre, from which all the lines are drawn, as in the third book it doth appear.   diagram  diagram 
Parallels, or gomowe lines be such lines as be drawn forth still in one distance, 
Parallelys Gemowe lines.  and are no nearer in one place then in an other, for and if they be nearer at one end then at the other, then are they no parallels, but may be called bought lines, and lo here examples of them both.   diagram 
tortuous parallels.   diagram 
parallelis.  
parallelis: circular.  
Concentrikes.   diagram 
bought lines  

A twine line.  And to return to my matter. an other fashioned line is there, which is named a twine or twist line, and it goeth as a wreyth about some other body. 
A spiral line. A worm line.  And an other sort of lines is there, that is called a spiral line, or a worm line, which representeth anapparant form of many circles, where there is not one in deed: of these ii kinds of lines, these be examples.   diagram 
A twist line.   diagram 
A spiral line   diagram 
A touch line.  
And when that a line doth cross the edge of the circle, 
A cord.  them is it called a cord, as you shall see anon in the speaking of circles.   diagram 
Match corner.  
March corner.  
Where A. and B. are match corners, so are C. and D. but not A. and C. neither D. and A.  
Now will I begin to speak of figures, that be properly so called, of which all be made of diverse lines, except only a circle, an egg form, and a tun form, which three have no angle, and have but one line for their bound, and an eye form which is made of one line, and hath an angle only.  
A circle is a figure made and enclosed with one line, 
A circle.  and hath in the middle of it a prick or centre, from which all the lines that be drawn to the circumfernece are equal all in length, as here you see.   diagram 

A diameter  And all the lines that be drawn cross the circle, and go by the centre, are named diameters, whose half, I mean from the centre to the circumference any way, 
Semidiameter.  is called the semidiameter, or half diameter.   diagram  diagram  diagram  diagram 
An arch.   diagram 
An egg form.   diagram 
A tun form.  

A tun or bar form  For if it be like the figure of a circle pressed in length, and both ends like big, then is it called a tun form, or barrel form, the right making of which figures, I will declare hereafter in the third book.  
another form there is, which you may call a nut form, and is made of one line much like an egg form, save that it hath a sharp angle.  
And it chanceth sometime that there is a right line drawn cross these figures, and that is called an axelyne, 
An axtre or axe line.  or axtre. Howbeit properly that line that is called an axtre, which goeth thorough the middle of a Globe, for as a diameter is in a circle, so is an axe line or axtre in a Globe, that line that goeth from side to side, and passeth by the middle of it. And the two points that such a line maketh in the utter bound or plat of the globe, are named polis, which you may call aptly in english, turn points: of which I do more largely entreat, in the book that I have written of the use of the globe.   diagram  diagram  diagram 
An other hath two compassed lines and one right line, and is as the portion of half a globe, example of B.   diagram  diagram  diagram  diagram  diagram  diagram  diagram  diagram  diagram  diagram  diagram  diagram  diagram  diagram  diagram  diagram  diagram  diagram  diagram  diagram 
And thus I make an eand to speak of plat forms, and will briefly say somewhat touching the figures of bode is which partly have one plat form fortheir bound, and that just round as a globe hath, or ended long as in an egg, and a tun  diagram 
The globe as is before  form, whose pictures are these.  
How be it you must mark that I mean not the very figure of a tun, when I say tun form, but a figure like a tun, for a tune form, hath but one plat form, and therefore must needs be round at the ends, where as a tun hath three plat forms, and is flat at each end, as partly these pictures do show.   diagram  diagram  diagram 
But as these forms be hard to be judged by their pycturs, so I do intend to pass them over with a great number of other forms of bodies, which afterward shall be set forth in the book of Perspective, because that without perspective knowledge, it is not easy to judge truly the forms of them in flat protacture.  
And thus I make an end for this time, of the definitions Geometrical, appertaining to this part of practice, and the rest will I prosecute as cause shall serve.   

THE PRACTIC WORKING OF sundry conclusions Geometrical.  

THE first CONCLUSION.  
To make a threlike triangle or any line measurable.  
TAKE THE JUST length of the line with your compass, and stay the one foot of the compass in one of the ends of that line, turning the other up or down at your will, drawing the arch of a circle against the middle of the line, and do likewise with the same  diagram compass unaltered, at the other end of the line and where these ij. crooked lines doth cross, from thence draw a line to each end of your first line, and there shall appear a threlike triangle drawn on that line.  

Example.  
A.B. is the first line, on which I would make the threlike triangle, therefore I open the compass as wide as that line  diagram is long, and draw two arch lines that meet in C, then from C. I draw ij other lines one to A, another to B, and than I have my purpose.    

THE. II. CONCLUSION. If you will make a twileke or a novelike triangle on any certain line.  
Consider first the length that you will have the other sides to contain, and to that length open your compass, and then work as you did in the threleke triangle, remembering this, that in a novelike triangle you must take ij. lengths beside the first line, and draw an arch line with one of them at the one end, and with the other at the other end, the example is as in the other before.  diagram   

THE III. CONCL. To divide an angle of right lines into ij. equal parts.  
First open your compasseas largely as you can, so that it do not exceed the length of the shortest line that encloseth the angle. Then set one foot of the compass in the very point of the angle and with the other foot draw a compassed arch from the one line of the angle to the other,  diagram that arch shall you divide in half, and then draw a line from the angle to the middle of that arch, and so the angle is divided into ij. equal parts. Example. Let the triangle be A. B.C, then set I one foot of the compass in B, and with the other I draw the arch D. E, which I part into ij. equal parts in F, and then draw a line from B, to F, & so I have mine intent   

THE FOUR CONCL. To divide any measurable line into ij. equal parts.  
 diagram  
Open your compass to the just length of the line. And then set one foot steddely at the one end of the line, & with the other foot draw an arch of a circle against the middle of the line, both over it, and also under it, then do like waise at the other end of the line. And mark where those arch lines do meet crossways, and between those ij. pricks draw a line, and it shall cut the first line in two equal portions.  

Example.  
The line is A. B. according to which I open the compass and make four arch lines, which meet in C. and D, then draw I a line from C, so have I my purpose.  
This conclusion serveth for making of quadrates and squires, beside many other commodities, howbeit it may be done more readylye by this conclusion that followeth next.    

THE FIFT CONCLUSION. To make a plum line or any prick that you will in any right line appointed.   diagram 
Example  
The line is A.B. the prick on which I should make the plum line, is C. then open I the compass as wide as A, C, and set one foot in C. and with the other do I mark out C.A. and C. B, then open I the compass as wide as A. B, and make ij. arch lines which do cross in D, and so have I done.  
How be it, it happeneth so sometimes, that the prick on which you would make the perpendicular or plum line, is so near the end of your line, that you can not extend any notable length from it to th'one end of the line, and if so be it then that you may not draw your line longer from that end, then doth this conclusion require a new aid, for the last devise will not serve. In such case therefore shall you do thus: If your line be of any notable length, divide it into five parts. And if it be not so long that it may yield five notable parts, then make an other line at will, and part it into five equal portions: so that three of those parts may be found in your line. Then open your compass as wide as three of these five measures be, and set the one foot of the compass in the prick, where you would have the plum line to light (which I call the first prick,) and with the other foot draw an arch line right over the prick, as you can aim it: then open your compass as wide as all five measures be, and set the one foot in the fourth prick, and with the other foot draw an other arch line cross the first, and where they two do cross, thence draw a line to the point where you would have the perpendicular line to light, and you have done.  

Example.  
 diagram The line is A. B. and A. is the prick, on which the perpendicular line must light. Therefore I divide A. B. into five parts equal, then do I open the compass to the wideness of three parts (that is A. D.) and let one foot stay in A. and with the other I make an arch line in C. afterward I open the compass as wide as A.B. (that is as wide as all five parts) and set one foot in the four prick, which is E, drawing an arch line with the other foot in C. also. Then do I draw thence a line unto A, and so have I done. But and if the line be to short to be parted into five parts, I shall divide it into iij. parts only, as you see the line F. G, and then make D. an other line (as is K. L.) which I divide into .v. such divisions, as F. G. containeth iij, then open I the compaas as wide as four parts (which is K. M.) and so set I one foot of the compass in F, and with the other I draw an arch line toward H, then open I the compass as wide as K. L. (that is all .v. parts) and set one foot in G, (that is the iij. prick) and with the other I draw an arch line toward H. also: and where those two arch lines do cross (which is by H.) thence draw I a line unto F, and that maketh a very plumb line to F. G, as my desire was. The manner of working of this conclusion, is like to the second conclusion, but the reason of it doth depend of the xlvi proposition of the first book of Euclid. another way yet. set one foot of the compass in the prick, on which you would have the plumb line to light, and stretch forth tother foot toward the longest end of the line, as wide as you can for the length of the line, and so draw a quarter of a compass or more, then without stirring of the compass, set one foot of it in the same line, where as the circularline did begin, and extend tother in the circular line, setting a mark where it doth light, then take half that quantity more thereunto, and by that prick that endeth the last part, draw a line to the prick assigned, and it shall be a perpendicular.   

Example.  
A. B. is the line appointed, to which  diagram I must make a perpendicular line to light in the prick assigned, which is A. Therefore do I set one foot of the compass in A, and extend the other unto D. making a part of a circle more than a quarter, that is D. E. Then do I set one foot of the compass unaltered in D, and stretch the other in the circular line, and it doth light in F, this space between D. and F. I divide into half in the prick G, which half I take with the compass, and set it beyond F. unto H, and therefore is H. the point, by which the perpendicular line must be drawn, so say I that the line H. A, is a plumb line to A. B, as the conclusion would.    

THE. VI CONCLUSION. To draw a straight line from any prick that is not in a line, and to make it perpendicular to an other line.  
Open your compass so wide that it may extend somewhat farther, than from the prick to the line, than set the one foot of  diagram the compass in the prick, and with the other shall you draw a compassed line, that shall cross that other first line in two places Now if you divide that arch line into two equal parts, and from the middle prick thereof unto the prick without the line you draw a straight line, it shallbe a plumb line to that first line, according to the conclusion.  

Example.  
C. is the appointed prick, from which unto the line A. B. I must draw a perpendicular. Therefore I open the compass so wide, that it may have one foot in C, and tother to reach over the line, and with that foot I draw an arch line as you see, between A. and B, which arch line I divide in the middle in the point D. Then draw I a line from C. to D, and it is perpendicular to the line A. B, according as my desire was.    

THE. VII. CONCLUSION. To make a plumb line or any portion of a circle, and that on the utter or inner bughte.  
Mark first the prick where the plumb line shall light: and prick out on each side of it two other points equally distant from that first prick. Then set the one foot of the compass in one of those side pricks, and the other foot in the other side prick, and first move one of the feet and draw an arch line over the middle prick, than set the compass steady with the one foot in the other side prick, and with the other foot draw an other arch line, that shall cut that first arch, and from the very point of their meeting, draw a right line unto the first prick, where you do mind that the plumb line shall light. And so have you performed th'intent of this conclusion.  

Example.  
The arch of the circle on which I would erect a plumb line, is A. B. C. and B. is the prick where I would have the  diagram plumb line to light. Therefore I meat out two equal distances on each side of that prick B. and they are A. C. Then open I the compass as wide as A. C. and setting one of the feet in A. with the other I draw an arch line which goeth by G. likeways I set one foot of the compass steadily in C. and with the other I draw an arch line, going by G. also▪ Now considering that G. is the prick of their meeting, it shall be also the point from which I must draw the plumb line. Then draw I aright line from G. to B. and so have mine intent. Now as A. B. C. hath a plumb line erected on his utter bought, so may I erect a plumb line on the inner bught of D. E. F, doing with it as I did with the other, that is to say, first setting for the the prick where the plumb line shall light, which is E, and then marking one other on each side, as are D. and F. And then proceeding as I did in the example before.    

THE VIII. conclusion. How to divide the arch of a circle into two equal parts, without measuring the arch.  
Divide the cord of that line into ij. equal portions, and then from the middle prick erect a plumb line, and it shall part that arch in the middle.  

Example.  
The arch to be divided is  diagram A. D.C, the cord is A, B. C, this cord is divided in the middle with B, from which prick if I erect a plum line as B. D, then will it divide the arch in the middle, that is to say, in D.    

THE IX. CONCLUSION.  
To do the same thing other wise. And for shortness of work, if you will make a plumb line without much labour, you may do it with your squire, so that it be justly made, for if you apply the edge of the squire to the line in which the prick is, and foresee the very corner of the squire do touch the prick. And than from that corner if you draw a line by the other edge of the squire, it will be a perpendicular to the former line.  

Example.  
A.B. is the line, on which I  diagram would make the plum line, or perpendicular. And therefore I mark the prick, from which the plumb line must rise, which here is C. Then do I set one edge of my squire (that is B.C.) to the line A. B, so that the corner of the squire do touch C. justly. And from C. I draw a line by the other edge of the squire, (which is C. D.) And so have I made the plum line D. C, which I sought for.    

THE X. CONCLUSION. How to do the same thing an other way yet  
If so be it that you have an arch of such greatness, that your squire will not suffice thereto, as the arch of a bridge or of a house or window, then may you do this. Meet underneath the arch where the middle of his cord will be, and there set a mark Then take a long line with a plummet, and hold the line in such a place of the arch, that the plummet do hang justly over the middle of the cord,  diagram that you did divide before, and then the line doth show you the middle of the arch.  

Example.  
The arch is A. D.B, of which I try the middle thus. I draw a cord from one side to the other (as here is A. B,) which I divide in the middle in C. Then take I a line with a plummet (that is D. E,) and so hold I the line that the plummet E, doth hang over C, And then I say that D. is the middle of the arch. And to thenien● that my plummet shall point the more justly, I do make it sharp at the neither end, and so may I trust this work for certain.    

THE XI. CONCLUSION. when any line is appointed and without it a prick, whereby a parallel must be drawn how you shall do it,  
Take the just measure between the line and the prick, according to which you shall open your compass. Then pitch one foot of your compass at the one end of the line, and with the other foot draw a bow line right over the pitch of the compass, likewise do at the other end of the line, then draw a line that shall touch the uttermost edge of both those bow lines, and it will be a true parallel to the first line appointed.  

Example.  
A. B, is the line unto which  diagram I must draw an other gemow line, which must pass by the prick C, first I meat with my compass the smallest distance that is from C. to the line, and that is C. F, wherefore staying the compass at that distance, I set the one foot in A, and with the other foot I make a bow line, which is D, then like wise set I the one foot of the compass in B, and with the other I make the second bow line, which is E. And then draw I a line, so that it toucheth the uttermost edge of both these bow lines, and that line passeth by the prick C, end is a gemowe line to A. B, as my seeking was.    

THE. XII. CONCLUSION. To make a triangle of any three lines, so that the lines be such, that any two of them be longer than the third. For this rule is general, that any two sides of every triangle taken together, are longer than the other side that remaineth.  
If you do remember the first and second conclusions, then is there no difficulty in this, for it is in manner the same work. First consiver the three lines that you must take, and set one of them for the ground line, then work with the other two lines as you did in the first and second conclusions.  

Example.  
 diagram I have three lines. A. B. and C. D. and E. F. of which I put. C.D. for my ground line, then with my compass I take the length of. A. B. and set the one foot of my compass in C, and draw an arch line with the other foot. likeways I take the length of E. F, and set one foot in D, and with the other foot I make an arch line cross the other arch, and the prick of their meeting (which is G.) shall be the third corner of the triangle, for in all such kinds of working to make a triangle, if you have one line drawn, there remaineth nothing else but to find where the pitch of the third corner shall be, for two of them must needs be at the two candes of the line that is drawn.    

THE XIII. CONCLUSION. If you have a line appointed, and a point in it limited, how you may make on it a right lined angle, equal to an other right lined angle, all ready assigned.  
first draw a line against the corner assigned, and so is it a triangle, then take heed to the line and the point in it assigned, and consider if that line from the prick to this end be as long as any of the sides that make the triangle assigned, and if it be long enough, then prick out there the length of one of the lines, and then work with the other two lines, according to the last conclusion, making a triangle of three like lines to that assigned triangle. If it be not long enough, thenne lengthen it first, and afterward do as I have said before.  

Example.  
Let the angle appointed  diagram be A. B. C, and the corner assigned, B. furthermore let the limited line be D. G, and the prick assigned D.  
first therefore by drawing the line A. C, I make the triangle A.B.C.  
Then considering that D. G, is longer than A. B, you shall cut out a line from D. toward  diagram G, equal to A. B, as for example D, F. Then measure out the other ij. lines and work with than according as the conclusion with the first also and the second teacheth you, and then have you done.    

THE XIIII. CONCLUSION. To make a square quadrate of any right line appointed.  
First make a plumb line unto your line appointed, which shall light at one of the ends of it, according to the fifth conclusion, and let it be of like length as your first line is, then open your compass to the just length of one of them, and set one foot of the compass in the end of the one line, and with the other foot draw an arch line, there as you think that the fourth corner shall beire after that set the one foot of the same compass unsturred, in the cande of the other line, and draw an other arch line cross the first arch line, and the point that they do cross in, is the prick of the fourth corner of the square quadrate which you seek for, therefore draw a line from that prick to the end of each line, and you shall thereby have made a square quadrate.  

Example.  
A. B. is the line  diagram proposed, of which I shall make a square quadrate, therefore first I make a plumb line unto it, which shall light in A, and that plumb line is A. B, then open I my compass as wide as the length of A. B, or B. C, (for they must be both equal) and I set the one foot of th'end in C, and with the other I make an arch line nigh unto D, afterward I set the compass again with one foot in B, A and with the other foot I make an arch line cross the first arch line in D, and from the prick of their crossing I draw two lines, one to B, A and an other to C, and so have I made the square quadrate that I intended.    

THE. XV. CONCLUSION. To make a likeiamme equal to a triangle appointed, and that in a right lined angle limited.  
First from one of the angles of the triangle, you shall draw a gemowe line, which shall be a parallel to that side of the triangle, on which you will make that likeiamme. Then on one end of the side of the triangle, which lieth against the gemowe line, you shall draw forth a line unto the gemow line, so that one angle that cometh of those two lines be like to the angle which is limited unto you. Then shall you divide into ij. equal parts that side of the triangle which beareth that line, and from the prick of that division, you shall raise an other line parallel to that former line, and continue it unto the first gemowe line, and then of those two last gemowe lines, and the first gemowe line, with the half side of the triangle, is made a lykeiamme equal to the triangle appointed, and hath an angle like to an angle limited, according to the conclusion.  
Example.  diagram B. C. G, is the triangle appointed unto, which I must make an equal likeiamme. And D, is the angle that the likeiamme must have. Therefore first intending to erect the likeiamme on the one side, that the ground line of the triangle (which is B. G.) I do draw a gemow line by C, and make it parallel to the ground line B. G, and that new gemow line is A. H. Then do I raise a line from B. unto the gemowe line, (which line is A. B) and make an angle equal to D, that is the appointed angle (according as the eight conclusion teacheth, and that angle is B. A. E. Then to proceed, I do part in the middle the said ground line B. G, in the prick F, from which prick I draw to the first gemowe line (A. H.) an other line that is parallale to A. B, and that line is E. F. Now say I that the likeiamme B. A. E. F, is equal to the triangle B. C. G. And also that it hath one angle (that is B. A. E. like to D. the angle that was limited. And so have I mine intent. The prose of the equalness of those two figures doth depend of the xli proposition of Euclides first book, and is the xxxi proposition of this second book of Theoremis, which saith, that when a triangle and a likeiamme be made between two self same gemow lines, and have their ground line of one length, then is the likeiamme double to the triangle, whereof it followeth, that if two such figures so drawn differ in their ground line only, so that the ground line of the likeiamme be but half the ground line of the triangle, then be those two figures equal, as you shall more at large perceive by the book of Theoremis, in the xxxi theorem.   

THE. XVI. CONCLUSION. To make a likeiamme equal to a triangle appointed, according to an angle limited, and on a line also assigned.  
In the last conclusion the sides of your likeiamme were left to your liberty, though you had an angle appointed. Now in this conclusion you are somewhat more restrained of liberty sith the line is limited, which must be the side of the likeiamme. Therefore thus shall you proceed. first according to the last conclusion, make a likeiamme in the angle appointed, equal to the triangle that is assigned. Then with your compass take the length of your line appointed, and set out two lines of the same length in the second gemowe lines, beginning at the one side of the likeiamme, and by those two pricks shall you draw an other gemowe line, which shall be parallel to two sides of the likeiamme. Afterward shall you draw two lines more for the accomplisment of your work, which better shall be perceived by a short example, then by a great numbered of words, only without example, therefore I will by example set forth the whole work.  

Example.  
first, according to the last  diagram conclusion, I make the likeiamme E. F. C. G, equal to the triangle D, in the appointed angle which is E. Then take I the length of the assigned line (which is A. B,) and with my compass I set forth the same length in the ij. gemow lines N. F. and H. G, setting one foot in E, and the other in N, and again setting one foot in C, and the other in H. Afterward I draw a line from N. to H, which is a gemow line, to ij. sides of the likeiamme. then draw I a line also from N. unto C, and extend it until it cross the lines, E. L. and F. G, which both must be drawn forth longer than the sides of the likeiamme. and where that line doth cross F. G, there I set M. Now to make an end, I make an other gemowe line, which is parallel to N. F. and H. G, and that gemowe line doth pass by the prick M, and then have I done. Now say I that H. C. K. L, is a likeiamme equal to the triangle appointed, which was D, and is made of a line assigned that is A. B, for H. C, is equal unto A. B, and so is K. L, The prose of the equalness of this likeiam unto the triangle, dependeth of the thirty and two Theorem: as in the hook of Theorems doth appear, where it is declared, that in all likeiams, when there are more than one made about one bias line, the filsquares of every of them must needs be equal.    

THE XVII. CONCLUSION. To make a likeiamme equal to any right lined figure, and that on an angle appointed.  
The readiest way to work this conclusion, is to turn that right lined figure into triangles, and then for every triangle together an equal likeiamme, according unto the eleven conclusion, and then to join all those likeiams into one, if their sides happen to be equal, which thing is ever certain, when all the triangles hap justly between one pair of gemow lines. but and if they will not frame so, then after that you have for the first triangle made his likeiamme, you shall take the length of one of his sides, and set that as a line assigned, on which you shall make all the other likeiams, according to the twelft conclusion,  diagram and so shall you have all your likeiams with ij. sides equal, and ij. like angles, so that you maieasily join them into one figure.  

Example.  
 diagram If the right lined figure be like unto A, then may it be turned into triangles that will stand between ij. parallels any ways, as you maise by C and D, for ij. sides of both  diagram the triangls are parallels. Also if the right lined figure be like unto E, then will it be turned into triangles, lying between two parallels also, as the other did before. as in the example of F. G. But and if the right lined figure be like unto H, and so turned into triangles as you see in K. L. M, where it is parted into iij triangles, them will not all those triangles lie between one pair of parallels or gemow lines, but must have many, for every triangle must have one pair of parallels several, yet it may happen that when there be three or sour triangles, ij. of them may happen to agree to one pair of parallels, which thing I remit to every honest wit to search, for the manner of their draft will declare, how many pairs of parallels they shall need, of which variety because the examples are infinite, I have set forth these few, that by them you may conjecture duly of all other like.  
 diagram  
Further explication you shall not greatly need, if you remember what hath been taught before, and then diligently behold how these sundry figures be turned into triangles. In the first you see I have made v. triangles, and four parallels. in the second seven. triangles and four parallels. in the third three triangles, and five parallels, in the iiij. you see five triangles & four parallels. in the fift, iiij. triangles and four parallels, & in the sixth theridamas are five triangles & iiij. parallels. Howbeit a man may at liberty alter them into divers forms of triangles & therefore I leave it to the discretion of the woorkmaister, to do in all such cases as he shall think best, for by these examples (if they be well marked) may all other like conclusions be wrought.    

THE XVIII. CONCLUSION. To part a line assigned after such a sort, that the square that is made of the whole line and one of his parts, shall be equal to the squar that cometh of the other part alone.  
First divide your line into ij. equal parts, and of the length of one part make a perpendicular to light at one end of your line assigned. then add a bias line, and make thereof a triangle, this done if you take from this bias line the half length ol your line appointed, which is the just length of your perpendicular, that part of the bias line which doth remain, is the greater portion of the division that you seek for, therefore if you cut your line according to the length of it, then will the square of that greater portior be equal to the square that is made of the whole line and his lesser portion. And contrary wise, the square of the whole line and his lesser part, will be equal to the square of the greater part.  

Example.  
A. B, is the line assigned. E. is the  diagram middle prick of A. B, B. C. is the plumb line or perpendicular, made of the half of A. B, equal to A. E, other B. E, the bias line is C. A, from which I cut a piece, that is C. D, equal to C. B, and according to the length loath piece that remaineth (which is D. A,) I do divide the line A. B, at which division I set F. Now say I, that this line A, B, (which was assigned unto me) is so divided in this point F, that that square of the hole line A. B, & of the one portion (that is F. B, the lesser part) is equal to the square of the other part, which is F. A, and is the greater part of the first line. The proof of this equality shall you learn by the xl Theorem.    

THE. XIX. CONCLUSION. To make a square quadrate equal to any right lined figure appointed.  
First make a likeiamme equal to that right lined figure, with a right angle, according to the xu conclusion, then consider the likeiamme, whether it have all his sides equal, or not: for if they be all equal, then have you done your conclusion. but and if the sides be not all equal, then shall you make one right line just as long as two of those unequal sides, that line shall you divide in the middle, and on that prick draw half a circle, then cut from that diameter of the half circle a certain portion equal to the one side of the likeiamme, and from that point of division shall you erect a perpendicular, which shall touch the edge of the circle. And that perpendicular shall be the just side of the square quadrate, equal both to the lykeiamme, and also to the right lined figure appointed, as the conclusion willed.  

Example.  
 diagram  
 diagram  
K, is the right lined figure appointed, and B. C. D. E, is the likeiamme, with right angles equal unto K, but because that this likeiamme is not a square quadrate, I must turn it into such one after this sort, I shall make one right line, as long as two unequal sides of the likeiamme, that line here is F. G, which is equal to B. C, and C. E. Then part I that line in the middle in the prick M, and on that prick I make half a circle, according to the length of the diameter F. G. Afterward I cut away a piece from F. G, equal to C. E, marking that point with H. And on that prick I erect a perpendicular H. K, which is the just side to the square quadrate that I seek for, therefore according to the doctrine of the ten conclusion, of that line I do make a square quadrate, and so have I attained the practice of this conclusion.    

THE. XX. CONCLUSION. when any two square quadrates are set forth, how you may make one equal to them both.  
First draw a right line equal to the side of one of the quadrates: and on the end of it make a perpendicular, equal in length to the side of the other quadrate, then draw a bias line between those two other lines, making thereof a right angeled triangle. And that bias line will make a square quadrate, equal to the other two quadrates appointed.  

Example.  
A.B. and C. D, are the two  diagram square quadrates appointed, unto which I must make one equal square quadrate. First therefore I do make a right line E. F, equal to one of the sides of the square quadrate A.B. And on the one end of it I make a plumb line E. G, equal to the side of the other quadrate D. C. Then draw I a bias line G. F, which being made the side of a quadrate (according to the tenth conclusion) will accomplish the work of this practice: for the quadrate H. is as much just as the other two. I mean A. B. and D. C.    

THE XXI. CONCLUSION. when any two quadrates be set forth, how to make a squire about the one quadrate, which shall be equal to the other quadrate.  
Determine with yourself about which quadrate you will make the squire, and draw one side of that quadrate forth in lengte, according to the measure of the side of the other quadrate, which line you may call the ground line, and then have you a right angle made on this line by an other side of the same quadrate: Therefore turn that into a right cornered triangle, according to the work in the last conclusion, by making of a bias line, and that bias line will perform the work of your desire. For if you take the length of that bias line with your compass, and then set one foot of the compass in the farthest angle of the first quadrate (which is the one end of the groundline) and extend the other foot on the same line, according to the measure of the bias line, and of that line make a quadrate, enclosing the first quadrate, then will there appear the form of a squire about the first quadrate, which squire is equal to the second quadrate.  
 diagram  

Example.  
The first square quadrate is A. B. C. D, and the second is E. Now would I maked squire about the quadrate A. B. C. D, which shall be equal unto the quadrate E.   diagram 
Therefore first I draw the line A. D, more in length, according to the measure of the side of E, as you see, from D. unto F, and so the hole line of both these several sides is A. F, them make I a bias line from C, to F, which bias line is the measure of this work. wherefore I open my compass according to the length of that bias line C. F, and set the one compass foot in A, and extend tother foot of the compass toward F, making this prick G, from which I erect a plumb line G. H, and so make out the square quadrate A. G. H. K, whose sides are equal each of them to A. G. And this square doth contain the first quadrate A. B. C. D, and also a squire G. H.K, which is equal to the second quadrate E, for as the last conclusion declareth, the quadrate A. G.H. K, is equal to both the other quadrates proposed, that is A. B. C.D, and E. Then must the squire G. H.K, needs be equal to E, considering that all the rest of that great quadrate is nothing else but the quadrate self, A. B. C. D, and so have I th'intent of this conclusion.    

THE. XXI. CONCLUSION. To find out the centre of any circle assigned.  
Draw a cord or string line cross the circle, then divide into two equal parts, both that cord, and also the bow line, or arch line, that serveth to that cord, and from the pricks of those divisions, if you draw an other line cross the circle, it must needs pass by the centre. Therefore divide that line in the middle, and that middle prick is the centre of the circle proposed.  

Example.  
Let the circle be A. B.C.D, whose centre I shall seek. First therefore I draw a cord cross the circle, that is A. C. Then do I divide that cord in the middle, in E, and likeways also do I divide his arch line A. B.C, in the middle, in the point B. Afterward I draw a line from B. to E, and so cross the circle, which line is B. D, in  diagram which line is the centre that I seek for. Therefore if I part that line B. D, in the middle in to two equal portions, that middle prick (which here is F) is the very centre of the said circle that I seek. This conclusion may other ways be wrought, as the most part of conclusions have sundry forms of practice, and that is, by making three pricks in the circumference of the circle, at liberty where you will, and then finding the centre to those three pricks, Which work because it serveth for sundry uses, I think meet to make it a several conclusion by itself.    

THE XXIII. CONCLUSION. To find the comen centre belonging to any three pricks appointed, if they be not in an exact right line.  
It is to be noted, that though every small arch of a great circle do seem to be a right line, yet in very deed it is not so, for every part of the circumference of all circles is compassed, though in little arches of great circles the eye cannot discern the crookedness, yet reason doth always declare it, therefore iij. pricks in an exact right line can not be brought into the circumference of a circle. But and if they be not in a right line how so ever they stand, thus shall you find their common centre. Open your compass so wide, that it be some what more than the half distance of two of those pricks. Then set the one foot of the compass in the one prick, and with the other foot draw an arch sign toward the other prick, Then again put the foot of your compass in the second prick, and with the other foot make an arch line, that may cross the first arch line in ij. places. Now as you have done with those two pricks, so do with the middle prick, and the third that remaineth. Then draw ij. lines by the points where those arch lines do cross, and where those two lines do meet, there is the centre that you seek for.  
Example   diagram  

THE XXIIII. CONCLUSION. To draw a touch line unto a circle, from any point assigned.  
Here must you understand that the prick must be without the circle, else the conclusion is not possible. But the prick or point being without the circle, thus shall you proceed: Open your compass, so that the one foot of it may be set in the centre of the circle, and the other foot on the prick appointed, and so draw an other circle of that largeness about the same centre: and it shall govern you certainly in making the said touch line. For if you draw a line from the prick appointed unto the centre of the circle, and mark the place where it doth cross the lesser circle, and from that point erect a plumb line that shall touch the edge of the utter circle, and mark also the place where that plumb line crosseth that utter circle, and from that place draw an other line to the centre, taking heed where it crosseth the lesser circle, if you draw a plumb line from that prick unto the edge of the greater circle, that line I say is a tooth line, drawn from the point assigned, according to the meaning of this conclusion.  

Example.  
Let the circle be called B.C. D, and his centre E, and the prick  diagram assigned A, open your compass now of such wideness, that the one foot may be set in E, which is the centre of the circle, & the other in A, which is the point assigned, & so make an other greater circle (as here is A. F G) them draw a line from A. unto E, and where that line doth cross the inner circle (which here is in the prick B.) there erect a plumb line unto the line. A.E. and let that plumb line touch the utter circle, as it doth here in the point F, so shall B.F. be that plumb line. Then from F. unto E. draw an other line which shall be F. E, and it will cut the inner circle, as it doth here in the point C, from which point C. if you erect a plumb line unto A, then is that line A. C, the touch line, which you should find. notwithstanding that this is a certain way to find any touch line, and a demonstrable form, yet more easily by many fold may you find and make any such line with a true ruler, laying the edge of the ruler to the edge of the circle and to the prick, and so drawing a right line, as this example showeth, where the circle is E, the prick assigned  diagram is A. and the ruler C. D. by which the touch line is drawn, and that is A, B, and as this way is light to do, so is it certain enough for any kind B of working.    

THE XXV. CONCLUSION. when you have any piece of the circumference of a circle assigned, how you may make out the whole circle agreeing thereunto.  
First seek out the centre of that arch, according to the do trine of the seventeenth conclusion, and then setting one foot of your compass in the centre, and extending the other foot un to the edge of the arch or piece of the circumference, it is easy to draw the whole circle.  

Example.  
A piece of an old pillar was found, like inform to this figure A.D.B. Now to know how much the compass of the hole pillar was, seeing by this part it appeareth that it was round, thus shall you do. Make in A table the like draft of that circumference by the self patron, using it as it were a crooked ruler  diagram Then make three pricks in that arch line, as I have made, C. D. and E. And then find out the common centre to them all, as the xvij conclusion teacheth. And that centre is here F, now setting one foot of your compass in F, and the other in C. D, other in E, and so making a compass, you have your whole intent.    

THE XXVI. CONCLUSION. To find the centre to any arch of a circle.  
If so be it that you desire to find the centre by any other way then by those three pricks, considering that sometimes you can not have so much space in the thing where the arch is drawn, as should serve to make those four bow lines, then shall you do thus: part that arch line into two parts, equal other unequal, it maketh no force, and unto each portion draw a cord, other a string line. And then according as you did in one arch in the xvi conclusion, so do in both those arches here, that is to say, divide the arch in the middle, and also the cord, and draw then a line by those two divisions, so then are you sure that that line goeth by the centre. Afterward do likeways with the other arch and his cord, and where those two lines do cross, there is the centre, that you seek for.  

Example.  
The arch of the circle is A. B. C, unto which I must seek a centre, therefore first I do  diagram divide it into two parts, the one of them is A. B, and the other is B. C. Then do I cut every arch in the middle, so is E. the middle of A. B, and G. is the middle of B.C. likeways, I take the middle of their cords, which I mark with F. and H, setting F. by E, and H. by G. Then draw I a line from E. to F, and from G. to H, and they do cross in D, wherefore say I, that D. is the centre, that I seek for.    

THE XXVII. CONCLUSION. To draw a circle within a triangle appointed.  
For this conclusion and all other like, you must understand, that when one figure is named to be within an other, that is not other ways to be understand, but that either every side of the inner figure doth touch every corner of the other, other else every corner of the one doth touch every side of the other. So I call that triangle drawn in a circle, whose corners do touch the circumference of the circle. And that circle is contained in a triangle, whose circumference doth touch justly every side of the triangle, and yet doth not cross over any side of it. And so that quadrate is called properly to be drawn in a circle, when all his four angles doth touch the edge of the circle, And that circle is drawn in a quadrate, whose circumference doth touch every side of the quadrate, and likeways of other figures.  
Examples are these. A. B. C. D. E. F.   diagram 
A. is a circle in a triangle.   diagram 
B. a triangle in a circle.   diagram 
C. a quadrate in a circle.   diagram 
D. a circle in a quadrate.   diagram  diagram 
In these two last figures E. and F, the circle is not named to be drawn in a triangle, because it doth not touch the sides of the triangle, neither is the triangle counted to be drawn in the circle, because one of his corners doth not touch the circumference of the circle, yet (as you see) the circle is within the triangle, and the triangle within the circle, but neither of them is properly named to be in the other. Now to come to the conclusion. If the triangle have all three sides like, then shall you take the middle of every side, and from the contrary corner draw a right line unto that point, and where those lines do cross one an other, there is the centre. Then set one foot of the compass in the centre, and stretch out the other to the middle prick of any of the sides, and so draw a compass, which shall touch every side of the triangle, but shall not pass with out any of them.  

Example.  
The triangle is A. B. C, whose sides I do part into two equal parts, each by itself in these points D. E. F, putting F. between A. B, and D. between B. C, and E. between A. C. Then draw I a line from C. to F, and an other from A. to D, and the third from B. to E.  
And where all those lines do meet (that is to say 〈◊〉 G,) I set the one foot of my compass,  diagram because it is the common centre, and so draw a circle according to the distance of any of the sides of the triangle. And then find I that circle to agree justly to all the sides of the triangle, so that the circle is justly made in the triangle, as the conclusion did purport. And this is ever true, when the triangle hath all three sides equal, other at the least two sides like long. But in the other kinds of triangles you must divide every angle in the middle, as the third conclusion teacheth you. And so draw lines from each angle to their middle prick. And where those lines do cross, there is the common centre, from which you shall draw a perpendicular to one of the sides. Then set one foot of the compass in that centre, and stretch the other foot according to the length  diagram of the perpendicular, and so draw your circle.   

Example.  
The triangle is A. B. C, whose corners I have divided in the middle with D. E. F, and have drawn the lines of division A. D. B. E. and C. F, which cross in G, therefore shall G. be the common centre. Then make I one perpendicular from G. unto the side A. C, and that is G.H. Now set I one foot of the compass in G, and extend the other foot unto H. and so draw a compass, which will justly answer to that triangle according to the meaning of the conclusion.    

THE XXVIII. CONCLUSION. To draw a circle about any triangle assigned.  
first divide two sides of the triangle equally in half, and from those ij. pricks erect two perpendiculars, which must needs meet in cross, and that point of their meeting is the centre of the circle that must be drawn, therefore set one foot of the compass in that point, and extend the other foot to one corner of the triangle, and so make a circle, and it shall touch all iij. corners of the triangle.  

Example.   diagram 
another way to do the same.  
And yet an other way may you do it, according as you learned in the seventeenth conclusion, for if you call the three corners of the triangle iij. pricks, and then (as you learned there) if you seek out the centre to those three pricks, and so make it a circle to enclose those three pricks in his circumference, you shall perceive that the same circle shall justly include the triangle proposed.   

Example.  
A. B. C. is the triangle, whose iij. corners I count to be iij. points. Then (as the seventeen conclusion doth teach) I  diagram seek a common centre, on which I may make a circle, that shall enclose those iij pricks that centre. as you see is D, for in D. doth the right lines, that pass by the angles of the arch lines, meet and cross. And on that centre as you see, have I made a circle, which doth enclose the iij. angles of the triangle, and consequent lie the triangle itself, as the conclusion did intend.    

THE XXIX. CONCLUSION. To make a triangle in a circle appointed whose corners shallbe equal to the corners of any triangle assigned.  
when I will draw a triangle in a circle appointed, so that the corners of that triangle shall be equal to the corners of any triangle assigned, then wust I first draw a touch line unto that circle, as the twenty conclusion doth teach, and in the very point of the touch must I make an angle, equal to one angle of the triangle, and that inward toward the circle: likewise in the same prick must I make an other angle with the other half of the touch line, equal to an other corner of the triangle appointed, and then between those two corners will there result a third angle, equal to the third corner of that triangle. Now where those two lines that enter into the circle, do touch the circumference (beside the touch line) there set I two pricks, and between them I draw a third line. And so have I made a triangle in a circle appointed, whose corners be equal to the corners of the triangle assigned.  

Example.  
A. B. C, is the triangle  diagram appointed, and F.G.H. is the circle, in which I must make an other triangle, with like  diagram angles to the angles of A.B.C. the triangle appointed. Therefore first I make the touch line D.F.E. And then make I an angle in F, equal to A, which is one of the angles of the triangle. And the line that maketh that angle with the touch line, is F. H, which I draw in length until it touch the edge of the circle. Then again in the same point F, I make an other corner equal to the angle C. and the line that maketh that corner with the touch line, is F.G. which also I draw forth until it touch the edge of the circle. And then have I made three angles upon that one touch line, and in that one point F, and those iij. angles be equal to the iij. angles of the triangle assigned, which thing doth plainly appear, in so much as they be equal to ij. right angles, as you may guess by the sixth theorem. And the three angles of every triangle are equal also to ij. right angles, as the two and twenty theorem doth show, so that because they be equal to one third thing, they must needs be equal together, as the common sentence saith. Then do I draw a line from G. to H, and that line maketh a triangle F.G.H. whose angles be equal to the angles of the triangle appointed. And this triangle is drawn in a circle, as the conclusion did will. The proof of this conclusion doth appear in the seventy and iiij. Theorem.    

THE XXX. CONCLUSION. To make a triangle about a circle assigned which shall have corners, equal to the corners of any triangle appointed.  
First draw forth in length the one side of the triangle assigned so that thereby you may have ij. utter angles, unto which two utter angles you shall make ij. other equal on the centre of the circle proposed, drawing three half diameters from the circumference, which shall enclose those ij. angles, them draw iij. touch lines which shall make ij. right angles, each of them with one of those semidiameters. Those iij. lines will make a triangle equally cornered to the triangle assigned, and that triangle is drawn about a circle appointed, as the conclusion did wil  

Example.  
A. B.C, is the triangle assigned, and G. H.K, is the circle appointed, about which I must make a triangle having equal angles to the angles of that triangle A.B.C. first therefore I draw A.C. (which is one of the sides of the triangle) in length that there may appear two utter angles in that triangle, as you see B. A. D, and B. C, E.  
Then draw I in the  diagram circle appointed a semidiameter, which is here H. F, for F. is the centre of the circle G. H.K. Then make I on that centre an angle equal  diagram to the utter angle B. A. D, and that angle is H.F. K. likeways on the same centre by drawing an other semidiameter, I make an other angle H. F. G, equal to the second utter angle of the triangle, which is B. C. E. And thus have I made three semidiameters in the circle appointed. Then at the end of each semidiameter, I draw a touch line, which shall make right angles with the semidiameter. And those three touch lines meet; as you see, and make the triangle L. M. N, which is the triangle that I should make, for it is drawn about a circle assigned, and hath corners equal to the corners of the triangle appointed, for the corner M. is equal to C. likeways L. to A, and N. to B, which thing you shall better perceive by the vi. Theorem, as I will declare in the book of proofs.    

THE XXXI. CONCLUSION. To make a portion of a circle on any right line assigned, which shall contain an angle equal to a right lined angle appointed.  
The angle appointed, may be a sharp angle, a right angle, other a blunt angle, so that the work must be diversely handled according to the diversities of the angles, but considering the hardness of those several works, I will omit them for a more meeter time, and at this time will she, we you one light way which serveth for all kinds of angles, and that is this. When the line is proposed, and the angle assigned, you shall join that line proposed so to the other two lines containing the angle assigned, that you shall make a triangle of them, for the easy doing whereof, you may enlarge or shorten as you see cause, nigh of the two lines containing the angle appointed. And when you have made a triangle of those iij. lines, then according to the doctrine of the seven and twety coclusion, make acircle about that triangle. And so have you wrought the request of this conclusion. which yet you may work by the twenty and eight conclusion also, so that of your line appointed, you make one side of the triangle be equal to  diagram the angleassigned as yourself may easily guess  

Example.  
First for example of a sharp angle let A. stand & B.C. shall be that line assigned. Then do I make a triangle, by adding B. C, as a third side to those other ij. which do include the angle assigned, and that triangle is D E. F, so that E. F. is the line appointed, and D. is the angle assigned. Then do I draw a portion of a circle about that triangle, from the one end of that line assigned unto the other, that is to say, from E. a long by D. unto F, which portion is evermore greater than the half of the circle, by reason that the angle is a sharp angle. But if the angle be right (as in the second example you see it) then shall the portion of the circle that containeth that angle, ever more be the just half of a circle. And when the angle is a blunt angle, as the third exaumpse doth propound, then shall the portion of the circle evermore be less than the half circle. So in the second example, G. is the right angle assigned, and H. K. is the line appointed, and L.M.N. the portion of the circle answering thereto. In the third example, O. is the blunt corner assigned, P. Q. is the line, and R. S. T. is the portion of the circle, that containeth that blunt corner, and is drawn on R. T. the line appointed.    

THE XXXII. CONCLUSION. To cut of from any circle appoineed, a portion containing an angle equal to a right lined angle assigned.  
When the angle and the circle are assigned, first draw a touch line unto that circle, and then draw an other line from the prick of the touching to one side of the circle, so that thereby those two lines do make an angle equal to the angle assigned. Then say I that the portion of the circle of the contrary side to the angle drawn, is the part that you scke for.  

Example.  
A. is the angle appointed, and D. E. F. is the circle assigned, from which I must cut away a portion that doth contain an angle equal to this angle A.  diagram Therefore first I do draw a touch line to the circle assigned, and that touch line is B. C, the very prick of the touch is D, from which D. J. draw a line D. E, so that the angle made of those two lines be equal to the angle appointed. Then say I, that the arch of the circle D. F. E, is the arch that I seek after. For if I do divide that arch in the middle (as here it is done in F.) and so draw thence two lines, one to A, and the other to E, then will the angle F, be equal to the angle assigned.    

THE XXXIII. CONCLUSION. To make a square quadrate in a circle assigned  
Draw two diameters in the circle, so that they run a cross, and that they make four right angles. Then draw four lines, that may join the four ends of those diameters, one to an other, and then have you made a square quadrate in the circle appointed.  

Example.  
A. B. C. D. is the circle assigned, and A. C. and B. D. are the  diagram two diameters which cross in the centre E, and make four right corners. Then do I make four other lines, that is A. B, B. C, C. D, and D. A, which do join together the four ends of the ij. diameters. And so is the square quadrate made in the circle assigned, as the conclusion willeth.    

THE XXXIIII. CONCLUSION. To make a square quadrate about any circled assigned.  
Draw two diameters in cross ways, so that they make four right angles in the centre. Then with your compass take the length of the half diameter, and set one foot of the compass in each end of those diameters, drawing two arch lines at every pitching of the compass, so shall you have viii. arch lines. Then if you mark the pricks wherein those arch lines do cross, and draw between those iiij. pricks iiij right lines, then have you made the square quadrate according to the request of the conclusion.  

Example.  
A.B.C. is the circle assigned  diagram in which first I draw two diameters, in cross ways, making iiij. right angles, and those ij. diameters are A.C. and B.D. Then set I my compass (which is opened according to the semidiameter of the said circle) fixing one foot in the end of every semidiameter, and draw with the other foot two arch lines, one on every side. As first, when I set the one foot in A, then with the other foot I do make two arch lines, one in E, and an other in F. Then set I the one foot of the compass in B, and draw two arch lines F. and G. Like wise setting the compass foot in C, I draw two other arch lines, G. and H, and on D. I make two other, H. and E. Then from the crossings of those eight arch lines I draw iiij. straight lines, that is to say, E. F, and F. G. also G. H, and H. E, which iiij. strait lines do make the square quadrate that I should draw about the circle assigned.    

THE XXXV. CONCLUSION. To draw a circle in any square quadrate appointed.  
first divide every side of the quadrate into two equal parts, and so draw two lines between each two contrary points, and where those two lines do cross, there is the centre of the circle. Then set the one foot of the compass in that point, and stretch forth the other foot, according to the length to half one of those lines, and so make a compass in the square quadrate assigned.  

Example.  
A, B.C.D. is the quadrate appointed, in which I must make a circle.  diagram Therefore first I do divide every side in ij. equal parts, and draw ij. lines across, between each ij. contrary pricks, as you see E. G, and F. H, which meet in K, and therefore shall K, be the centre of the circle. Then do I set one foot of the compass in K. and open the other as wide as K. E, and so draw a circle, which is made ancordinge to the conclusion.    

THE XXXVI. CONCLUSION. To draw a circle about a square quadrate.  
Draw ij. lines between the iiij. corners of the quadrate, and where they meet in cross, there is the centre of the circle that you seek for. Then set one foot of the compass in that centre, and extend the other foot unto one corner of the quadrate, and so may you draw a circle which shall justly enclose the quadrate proposed.  

Example.  
A. B. C. D. is the square quadrate proposed, about which I must make a circle. Therefore do I draw ij. lines cross the square quadrate from angle to angle,  diagram as you see A. C. & B. D. And where they ij. do cross (that is to say in E.) there set I the one foot of the compass as in the centre, and the other foot I do extend unto one angle of the quadrate, as for exampse to A, and so make a compass, which doth justly enclose the quadrate, according to the mind of the conclusion.    

THE XXXVII. CONCLUSION. To make a twileke triangle, which shall have every of the ij. angles that lie about the ground line, double to the other corner.  
first make a circle, and divide the circumference of it into five equal parts. And then draw from one prick (which you will) two lines to ij. other pricks, that is to say to the iij. and iiij. prick, counting that for the first, wherehence you drew both those lines, Then draw the third line to make a triangle with those other two, and you have done according to the conclusion, and have made a twelike triangle, whose ij. corners about the ground line, are each of them double to the other corner.  

Example.  
A, B. C. is the circle, which I  diagram have divided into five equal portions. And from one of the pricks (which is A,) I have drawn ij. lines, A. B. and A. C, which are drawn to the third and iiij. pricks. Then draw I the third line C. B. which is the ground line, and maketh the triangle, that I would have, for the angle C. is double to the angle A, and so is the angle B. also.    

THE XXXVII. CONCLUSION. To make a cinquangle of equal sides, and equal corners in any circle appointed.  
Divide the circle appointed into five equal parts, as you did in the last conclusion, and draw ij. lines from every prick to the other ij. that are next unto it. And so shall you make a cinquangle after the meaning of the conclusion.  

Example.  
You see here this circle A. B. C. D. E. divided into five equal portions. And from each prick ij. lines drawn to the other ij. next pricks, so from A. are drawn ij. lines, one to B, and the other to E, and so from C. one to B. and an other to D, and likewise of  diagram the rest. So that you have not only learned hereby how to make a sinkangle in any circle, but also how you shall make a like figure speedily, when and where you will, only drawing the circle for the intent, readylye to make the other figure (I mean the cinquangle) thereby.    

THE XXXIX. CONCLUSION. How to make a cinquangle of equal sides and equal angles about any circle appointed.  
Divide first the circle as you did in the last conclusion into five equal portions, and draw five semidiameters in the circle. Then make five touch lines, in such sort that every touch line make two right angles with one of the semidiameters. And those five touch lines will make a cinquangle of equal sides and equal angles.  

Example.   diagram 
A. B. C. D. E. is the circle appointed, which is divided into five equal parts. And unto every prick is drawn a semidiameter, as you see. Then do I make a touch line in the prick B, which is F. G, making ij. right angles with the semidiameter B, and like ways on C. is made G. H, on D. standeth H. K, and on E, is set K. L, so that of those .v. touch lines are made the .v. sides of a cinkeangle, according to the conclusion.  
another way.  
another way also may you draw a cinkeangle about a circle, drawing first a cinkeangle in the circle (which is an easy thing to do, by the doctrine of the xxxvij conclusion) and then drawing .v. touch lines which shall be just parallels to the .v. sides of the cinkeangle in the circle, forseeing that one of them do not cross overthwart an other and then have you done. The example of this (because it is easy) I leave to your own exercise.    

THE XL. CONCLUSION. To make a circle in any appointed cinkeangle of equal sides and equal corners.  
Draw a plumb line from any one corner of the cinkeangle, unto the middle of the side that lieth just against that angle. And do likeways in drawing an other line from some other corner, to the middle of the side that lieth against that corner also. And those two lines will meet in cross in the prick of their crossing, shall you judge the centre of the circle to be. Therefore set one foot of the compass in that prick, and extend the other to the end of the line that toucheth the middle of one side, which you list, and so draw a circle. And it shall be justly made in the cinkeangle, according to the conclusion.  

Example.  
The cinkeangle assigned is A. B. C. D. E, in which I must make a circle, wherefore I draw a right line from the one angle (as from B,) to the middle of the contrary side (which is E. D,) and that middle prick is F. Then likeways from an other corner (as from E) I draw a right line to the middle of the side that lieth against it (which is B. C.) and that prick is  diagram G. Now because that these two lines do cross in H, I say that H. is the centre of the circle, which I would make. Therefore I set one foot of the compass in H, and extend the other foot unto G, or F. (which are the ends of the lines that light in the middle of the side of that cinkeangle) and so make I a circle in the cinquangle, right as the conclusion meaneth.    

THE XLI. CONCLUSION. To make a circle about any assigned cinkeangle of equal sides, and equal corners.  
Draw two lines within the cinkeangle, from two corners to the middle on the two contrary sides (as the last conclusion teacheth) and the point of their crossing shall be the centre of the circle that I seek for. Then set I one foot of the compass in that centre, and the other foot I extend to one of the angles of the cinquangle, and so draw I a circle about the cinkangle assigned.  

Example.   diagram 
another way also.  
another way may I do it, thus presupposing any three corners of the cinquangle to be three pricks appointed, unto which I should find the centre, and then drawing a circle touching them all three, according to the doctrine of the seventeen, one and twenty, and two and twenty conclusions. And when I have found the centre, then do I draw the circle as the samc conclusions do teach, and this forty conclusion also.    

THE XLII. CONCLUSION. To make a siseangle of equal sides, and equal angles, in any circle assigned.  
If the centre of the circle be not known, then seek out the centre according to the doctrine of the sixetenth conclusion. And with your compass take the quantity of the semidiameter justly. And then set one foot in one prick of the circumference of the circle, and with the other make a mark in the circumference also toward both sides. Then set one foot of the compass steadyly in each of those new pricks, and point out two other pricks. And if you have done well, you shall peceave that there will be but even six such divisions in the circumference. Whereby it doth well appear, that the side of any siseangle made in a circle, is equal to the semidiameter of the same circle.  

Example.  
 diagram  
The circle is B. C. D. E. F. G, whose centre I find to be A. Therefore I set one foot of the compass in A, and do extend the other foot to B, thereby taking the semidiameter. Then set I one foot of the compass unremoved in B, and mark with the other foot on each side C. and G. Then from C. I mark D, and from D, E: from E. mark I F. And then have I but one space just unto G. and so have I made a just siseangle of equal sides and equal angles, in a circle appointed.    

THE XLIII. CONCLUSION.  
To make a siseangle of equal sides, and equal angles about any circle assigned.   

THE XLIIII. CONCLUSION.  
To make a circle in any siseangle appointed, of equal sides and equal angles.   

THE XLV. CONCLUSION.  
To make a circle about any siseangle limited of equal sides and equal angles,  
Because you may easily conjecture the making of these figures by that that is said before of cinkangles, only considering that there is a difference in the numbered of the sides. I thought best to leave these unto your own device, that you should study in some things to exercise your wit withal and that you might have the better occasion to perceive what difference there is between each two of those conclusions. For though it seem one thing to make a siseangle in a circ e, and to make a circle about a siseangle, yet shall you perceive, that it is not one thing, neither are those two conclusions wrought one way. Like waise shall you think of those other two conclusions. To make a siseangle about a circle, and to make a circle in a siseangle, though the figures be one in fashion, when they are made, yet are they not one in working, as you may well perceive by the thirty seven. thirty viii xxxix. and xl. conclusions, in which the same works are taught, touching a circle and a cinquangle, yet this much will I say, for your help in working, that when you shall seek the centre in a size angle (whether it be to make a circled in it other about it) you shall draw the two cross lines, from one angle to the other angle that lieth against it, and not to the middle of any side, as you did in the cinquangle.   

THE XLVI. CONCLUSION.  
To make a figure of fifteen equal sides and angles in any circle appointed.  
This rule is general, that how many sides the figure shall have, that shall be drawn in any circle, into so many parts justly must the circles be divided. And therefore it is the more easier work commonly, to draw a figure in a circle, then to make a circle in an other figure. Now therefore to end this conclusion, divide the circle first into five parts, and then each of them into three parts again: Or else first divide it into three parts, and then each of them into five other parts, as you list, and can most readily. Then draw lines between every two pricks that be nighest together, and there will appear rightly drawn the figure, of fifteen sides, and angles equal. And so do with any other figure of what numbered of sides so ever it be.  
FINIS.    

THE SECOND BOOK OF THE PRINCIPLES of Geometry, containing certain Theorems, which may be called Approved truths. And be as it were the most certain grounds, whereon the practic conclusions of Geometry are founded.  
Whereunto are annexed certain declarations by examples, for the right understanding of the same, to the end that the simple reader might not justly complain of hardness or obscurity, and for the same cause are the demonstrations and just profess omitted, until a more convient time.  
1551.   

If truth may try itself,  
By Reason's prudent skill,  
If reason may prevail by right,  
And rule the rage of will,  
I dare the trial bide,  
For truth that I pretend.  
And though some list at me repine,  
just truth shall me defend.   

THE PREFACE UNTO the Theorems.  
I Doubt not gentle reader, but as my argument is strange and unacquainted with the vulgar tongue, so shall I of many men be strangely talked of, and as straunglye judged. Some men will say peradventure, I might have better employed my time in some pleasant history, comprising matter of chivasrye. Some other would more have praised my travail, if I had spent the like time in some moral matter, other in decising some controversy of religion. And yet some men (as I judge) will not mislike this kind of matter, but then will they wish that I had used a more certain order in placing both the Propositions and Theorems, and also a more exacter proof of each of them both, by demonstrations mathematical. Some also will mislike my shortness and simple plainness, as other of other affections diversely shall espy somewhat that they shall think blame worthy, and shall miss somewhat, that they would wish to have been here used, so that every man shall give his verdict of me according to his fantasy, unto whom jointly, I make this my first answer: that as they are many and in opinions very divers, so were it scarce possible to please them all with any one argument, of what kind so ever it were. And for my second answer, I say thus. That if any one argument might please them all, then should they be thankful unto me for this kind of matter. For neither is there any matter more strange in the english tongue, than this whereof never book was written before now, in that tongue, and therefore ought to delight all them, that desire to understand strange matters, as most men commonly do. And again the practice is so pleasant in using, and so profitable in applying, that who so ever do the delight in any of both, ought not of right to mislike this art. And if any man shall like the art well for itself, but shall mislike the form that I have used in teaching of it, to him I shall say, first, that I do wish with him that some other man, which could better have done it, had showed his good will, and used his diligence in such sort, that I might have been thereby occasioned justly to have left of my labour, or after my travail to have suppressed my books. But sith no man hath yet attempted the like, as far as I can learn, I trust all such as be not exercised in the study of geometry, shall find great ease and furtherance by this simple, plain, and easy form of writing. And shall perceive the exact works of Theon, and others that writ on Euclid, a great deal the sumner, by this blunt delineation afore hand to them taught, For I dare presuppose of them, that thing which I have set in myself, and have marked in others, that is to say, that it is not easy for a man that shall travail in a strange art, to understand at the beginning both the thing that is taught and also the just reason why it is so. And by experience of teaching I have tried it to be true, for when I have taught the proposition, as it imported in meaning, and annexed the demonstration with all, I did perceive that it was a great trouble and a painful vexation of mind to the learner, to comprehend both those things at ones. And therefore did I prove first to make them to understand the sense of the propositions, and then afterward did they conceive the demonstrations much sooner, when they had the sentence of the propositions first engrafted in their minds. This thing caused me in both these books to omit the demonstrations, and to use only a plain form of declaration, which might best serve for the first introduction. Which example hat been used by other learned men before now, for not only Georgius joachimus Rheticus, but also Boetius that witty clerk did set forth some whole books of Euclid, without any demonstration or any other declaration at al. But & if I shall hereafter perceive that it may be a thankful travail to set forth the propositions of geometry with demonstrations, I will not refuse to do it, and that with sundry varietees of demonstrations, both pleasant and profitable also. And then will I in like manner prepare to set forth the other books, which now are left unprinted, by occasion not so much of the charges in cutting of the figures, as for other just hinderances, which I trust hereafter shall be remedied. In the mean season if any man muse why I have set the Conclusions before the Teoremes, saying many of the Theorems seem to include the cause of some of the conclusions, and therefore ought to have gone before them, as the cause goeth before the effect. Hereunto I say, that although the cause do go before the effect in order of nature, yet in order of teaching the effect must be first declared, and than the cause thereof showed, for so shall men best understand things First to learn that such things are to be wrought, and secondarily what they are, and what they do import, and than thirdly what is the cause thereof. another cause why that the theorems be put after the conclusions is this, when I wrote these first cunclusions (which was. iiiij. years passed) I thought not then to have added any theorems, but next unto the conclusions to have taught the order how to have applied them to work, for drawing of plots, & such like uses. But afterward considering the great commodity that they serve for, and the light that they do give to all sorts of practice geometrical, beside other more notable benefits, which shall be declared more specially in a place convenient, I thought best to give you some taste of them, and the pleasant contemplation of such geometrical propositions, which might serve diversely in other books for the demonstrations and proofs of all Geometrical works. And in them, as well as in the propositions, I have drawn in the Linearie examples many times more lines, than be spoken of in the explication of them, which is done to this intent, that if any man list to learn the demonstrations by heart, as some learned men have judged best to do) those same men should find the Linearye examples to serve for this purpose, and to want no thing needful to the just proof, whereby this book may be well approved to be more complete than many men would suppose it.  
And thus for this time I will make an end without any larger declaration of the commodities of this art, or any farther answering to that may be objected against my handling of it, willing them that mislike it, not to meddle with it: and unto those that will not disdain the study of it, I promise all such aid as I shall be able to show for their farther proceeding both in the same, and in all other commodities that thereof may ensue. And for their encouragement I have here annexed the names and brief arguments of such books, as I intend (God willing) shortly to set forth, if I shall perceive that my pains may profit other, as my desire is.   

The brief arguments of such books as are appointed shortly to be set forth by the author hereof.  
THE second part of Arithmetic, teaching the working by fractions, with extraction of roots both square and cubike: And declaring the rule of allegation, with sundry pleasant examples in metals and other things. Also the rule of false position, with divers examples not only vulgar, but some appertaining to the rule of Algeber, applied unto quantitees partly rational, and partly surde.  
THE art of Measuring by the quadrate geometrical, and the disorders committed in using the same, not only revealed but reform also (as much as to the instrument pertaineth) by the devise of a new quadrate newly invented by the author hereof.  
THE art of measuring by the astronomers staff, and by the astronomers ring, and the form of making them both.  
THE art of making of Dial's, both for the day and the night, with certain new forms of fixed dials for the moon and other for the stars, which may be set in glass windows, to serve by day and by night. And how you may by those dials know in what degree of the Zodiac not only the son, but also the moon is. And how many hours old she is. And also by the same dial to know whether any eclipse shall be that month, of the son or of the moon.  
The making and use of an instrument, whereby you may not only measure the distance at ones of all places that you can see together, how much each one is from you, and every one from other, but also thereby to draw the plot of any country that you shall come in, as justly as may be, by man's diligence and labour.  
THE use both of the Globe and the Sphere, and therein also of the art of Navigation, and what instruments serve best thereunto, and of the true latitude and longitude of regions and towns.  
Euclides works in four parts, with divers demonstrations Arithmetical and Geometrical or Linearie. The first part of plat forms. The second of numbers and quantitees surde or irrational. The third of bodies and  forms. The fourth of perspective, and other things thereto annexed.  
BESIDE these I have other sundry works partly ended, And partly to be ended, Of the peregrination of man, and the original of all nations, The state of times, and mutations of reasmes, The image of a perfect common wealth, with divers other works in natural sciences, Of the wonderful works and effects in beasts, plants, and minerals, of which at this time, I will omit the arguments, because they do appertain little to this art, and handle other matters in an other sort.  

To have, or leave,  
Now may you choose,  
No pain to please,  
Will I refuse.    

The Theorems of Geometry, before WHICH ARE SET FORTH certain grauntable requests which serve for demonstrations Mathematical.  

That from any prick to one other, there may be drawn a right line.  
AS for example  diagram A being the one prick, and B. the other, you may draw between them from the one to the other, that is to say, from A. unto B, and from B. to A.   

That any right line of measurable length may be drawn forth longer, and strait.  
Example of A. B, which as it is  diagram a line of measurable length, so may it be drawn forth farther, as for example unto C, and that in true streightenes without crokinge.   

That upon any centre, there  diagram may be made a circle of any quantity that a man will.  
Let the centre be set to be A, what shall hinder a man to draw a circle about it, of what quantity that he lusteth, as you see the form here: other bigger or less, as it shall like him to do?   

That all right angles be equal each to other.  
Set for an example A. and B, of which two  diagram though A. seem the greater angle to some men of small experience, it happeneth only because that the, lines about A, are longer than the lines about B, as you may prove by drawing them longer, for so shall B. seem the greater angle▪ if you make his lines longer than the lines that make the angle A. And to prove it by demonstration, I say thus. If any ij. right corners be not equal, than one right corner is greater than an other, but that corner which is greater than a right angle, is a blunt corner (by his definition) so must one corner be both a right corner and a blunt corner also, which is not possible: And again: the lesser right corner must be a sharp corner, by his definition, because it is less than a right angle. which thing is impossible. Therefore I conclude that all right angles be equal.   

If one right line do cross two other right lines, and make ij. inner corners of one side lesser than ij. right corners, it is certain, that if those two lines be drawn forth right on that side that the sharp inner corners be, they will at length meet together, and cross on an other.  
The ij. lines being as  diagram A. B. and C. D, and the third line crossing them as doth here E. F, making ij inner corns (as are G.H.) lesser than two right corners, fithech of them is less than a right corner, as your eyes may judge, then say I, if those ij. lines A.B. and C.D. be drawn in length on that side that G. and H. are, the will at length meet and cross one an other.   

Two right lines make no plat form.  
A plat form, as you hard before, hath both length and hredthe, and is enclosed with lines as with his bounds, but ij. right lines cannot enclose all the bonds  diagram of any plat form. Take for an example first these two right lines A B. and A. C, which meet together in A, but yet cannot be called a plat form, because there is no bond from B. to C, but if you will draw a line between them two, that, is from B. to C, then will it be a plat form, that is to say, a triangle, but then are there iij. lines, and not only ij. Likewise may you say of D.E. and F. G, which do make a plat form, neither yet can they make any without help of two lines more, whereof the one must be drawn from D. to F, and the other from E. to G, and then will it be a long square. So then of two right lines can be made no plat form. But of ij. crooked lines be made a plat form, as you see in the eye form. And also of one right line, & one crooked line, may a plat form be made, as the semicircle F. doothesette forth.   

Certain common sentences manifest to sense, and acknowledged of all men.  

The first common sentence. What so ever things be equal to one other thing, those same be equal between themselves.  
Examples thereof you may take both in greatness and also in numbered. First (though it pertain not properly to geometry, but to help the understanding of the rules, which may be wrought by both arts) thus may you perceive. If the sum of money in my purse, and the money in your purse be equal each of them to the money that any other man hath, then must needs your money  diagram and mine be equal together. Likewise, if any ij. quantities, as A and B, be equal to an other, as unto C, then must needs A. and B. be equal each to other, as A. equal to B, and B. equal to A, which thing the better to peceave, turn these quantities into numbered, so shall A. and B. make fixteene, and C. as many. As you may perceive by multiplying the numbered of their sides together.   

The second common sentence. And if you add equal portions to things that be equal, what so amounteth of them shallbe equal.  
Example, If you and I have like sums of money, and then receive each of us like sums more, than our sums will be like still. Also if A. and B. (as in the former example) be equal, then by adding an equal portion to them both, as to each of them, the quarter of A. (that is four) they will be equal still.   

The third common sentence. And if you abate even portions from things that are equal, those parts that remain shall be equal also.  
This you may perceive by the last example. For that that was added there, is subtracted here. and so the one doth approve the other.   

The fourth common sentence. If you abate equal parts from unequal things, the remainers shall be unequal.  
As because that a hundredth and eight and forty be unequal if I take ten from them both, there will remain ninety and eight and thirty, which are also unequal. and likewise in quantities it is to be judged.   

The fift common sentence. when even portions are added to unequalle things, those that amount shallbe unequal.  
So if you add twenty to fifty, and like ways to ninety, you shall make seventy. and a hundred and ten which are no less unequal, than were fifty and ninety.   

The sixth common sentence. If two things be double to any other, those same two things are equal together.  
 diagram  
Because A. and B. are each of them double to C, therefore must A. and B. needs be equal together. For as v. times viii. maketh xl. which is double to iiij. times v, that is xx, so iiij. times x, likewise is double to xx. (for it maketh forty) and therefore must needs be equal to forty.   

The seventh common sentence. If any two things be the halves of one other thing, than are they two equal together.  
So are D. and C. in the last example equal together, because they are each of them the half of A. other of B, as their numbered declareth.   

The eight common sentence. If any one quantity be laid on an other, and they agree, so that the one exceedeth not the other, then are they equal together.  
As if this figure A. B. C, be laid on  diagram that other D. E. F, so that A. be laid to D, B. to E, and C. to F, you shall see them agree in sides exactly and the one not to exceed the other, for the line A.B. is equal to D. E, and the third line C. A, is equal to F. D so that every side in the one is equal to some one side of the other. wherefore it is plain, that the two triangles are equal together.   

The ninth common sentence. Every whole thing is greater than any of his parts.  
This sentence needeth none example. For the thing is more plainer than any declaration, yet considering that other comen sentence that followeth next that.   

The tenth common sentence. Every whole thing is equal to all his parts taken together.  
It shall be meet to express both with one example, for of this last sentence many men at the first hearing do make a doubt. Therefore as in this example of the circle divided into sundry parts  diagram it doth appear that no part can be so great as the whole circle, (according to the meaning of the eight sentence) so yet it is certain, that all those eight parts together be equal unto the whole circle. And this is the meaning of that common sentence (which many use, and few do rightly understand) that is, that All the parts of any thing are nothing else, but the whole. And contrary ways: The whole is nothing else, but all his parts taken together. which sayings some have understand to mean thus: that all the parts are of the same kind that the whole thing is: but that that meaning is false, it doth plainly appear by this figure  diagram A. B, whose parts A. and B, are triangles, and the whole figure is a square, and so are they not of one kind. But and if they apply it to the matter or substance of things (as some do) then is it most false, for every compound thing is made of parts of diverse matter and substance. Take for example a man, a house, a book, and all other compound things. Some understand it thus, that the parts all together can make none other form, but that that the whole doth show, which is also false, for I may make five hundred diverse figures of the parts of some one figure, as you shall better perceive in the third book. And in the mean season take for an example this square figure following A. B. C. D, which is divided but into two parts, and yet (as youse) I have made five figures more beside the first, with only diverse joining of those two parts. But of this shall I speak more largely in an other place, in the mean season content yourself with these principles, which are certain of the chief grounds whereon all demonstrations mathematical are formed, of which though the most part seem so plain, that no child doth doubt of them, think not therefore that the art unto which they serve, is simple, other childish, but rather consider, how certain the profess of that art is, that  diagram hath for his grounds such plain truths, & as I may say, such undowbtfull and sensible principles, And this is the cause why all learned men doth approve the certainty of geometry, and consequently of the other arts mathematical, which have the grounds (as Arithmetic, music and astronomy) above all other arts and sciences, that be used amongst men. Thus much have I said of the first principles, and now will I go on with the theorems, which I do only by examples declae, minding to reserve the proofs to a peculiar book which I will then set forth, when I perceive this to be thankfully taken of the readers of it.    

The theorems of Geometry briefly declared by short examples.  

The first Theorem. When two triangles be so drawn, that the one of them hath ij. sides equal to ij. sides of the other triangle, and that the angles enclosed with those sides, be equal also in both triangles, then is the third side likewise equal in them. And the whole triangles be of one greatness, and every angle in the one equal to his match angle in the other, I mean those angles that be enclosed with like sides.  

Example.  
This triangle A.B.C. hath ij. sides (that is to say) C.A. and  diagram C. B, equal to ij. sides of the other triangle F. G.H, for A. C. is equal to F. G, and B.C. is equal to G.H. And also the angle C. contained between F. G, and G. H, for both of them answer to the  diagram eight part of a circle. Therefore doth it remain that A. B. which is the third line in the first triangle, doth agree in length with F. H, which is the third line in that second triangle & the hole triangle. A.B.C. must needs be equal to the hole triangle F.G.H. And every corner equal to his match, that is to say, A. equal to F, B. to H, and C. to G, for those be called match corners, which are enclosed with like sides, other else do lie against like sides.    

The second Theorem. In twileke triangles the ij. corners that be about the ground line, are equal together. And if the sides that be equal, be drawn out in length them will the corners that are under the ground line, be equal also together.  

Example  
A.B.C. is a twileke triangle, for  diagram the one side A. C, is equal to the other side B.C. And therefore I say that the inner corners A. and B, which are about the ground lines, (that is A.B.) be equal together. And farther if C. A. and C. B. be drawn forth unto D and E. as you see that I have drawn them, then say I that the two utter angles under A. and B, are equal also together: as the theorem said. The proof whereof, as of all the rest, shall appear in Euclid, whom I intend to set forth in english with sundry new additions, if I may perceive that it willbe thankfully taken.    

The third Theorem. If in any triangle there be two angles equal together, then shall the sides, that lie against those angles, be equal also.  

Example  
This triangle A.B.C. hath two corners equal  diagram each to other, that is A. and B, as I do by supposition limit, wherefore it followeth that the side A. C, is equal to that other side B. C, for the side A. C, lieth against the angle B, and the side B. C, lieth against the angle A.    

The fourth Theorem. when two lines are drawn from the ends of any one line, and meet in any point, it is not possible to draw two other lines of like length each to his match that shall begin at the same points, and end in any other point than the two first did.  

Example.  
The first line is A. B, on which I have  diagram erected two other lines A. C, and B. C, that meet in the prick C, wherefore I say, it is not possible to draw ij. other lines from A. and B. which shall meet in one point (as you see A. D. and B.D. meet in D.) but that the match lines shallbe unequal, I mean by match lines, the two lines on one side that is the ij. on the right hand, or the ij. on the left hand, for as youse in this example A. D. is longer than A. C, and B.C. is longer than B.D. And it is not possible, that A. C. and A. D. shall be of one length, if B. D. and B.C. be like long. For if one couple of match lines be equal (as the same example A.E. is equal to A.C. in length) then must B. E needs be unequal to B.C. as you see, it is here shorter.    

The fift Theorem. If two triangles have there ij. sides equal one to another, and their ground lines equal also, then shall their corners, which are contained between like sides, be equal one to the other.  

Example.  
Because these two triangles A.B. C, and D.E.F. have two sides equal one to an other.  diagram For A. C. is equal to D. F, and B.C. is equal to E. F, and again their ground lines A.B. and D.E. are like in length, therefore is each angle of the one triangle equal to each angle of the other, comparing together those angles that are contained within like sides, so is A. equal to D, B. to E, and C. to F, for they are contained within like sides, as before is said.    

The sixth Theorem. when any right line standeth on an other, the ij. angles that they make, other are both right angles, or else equal to two right angles.  

Example.  
 diagram A.B. is a right line, and on it there doth light another right line, drawn from C. perpendicularly on it, therefore say I, that the two angles that they do make, are two right angles as may be judged by the definition of a right angle. But in the second part of the example, where A.B. being still the right line, on which D. standeth in slope ways, the two angles that be made of them are not right angles, but yet they are equal to two right angles, for so much as the one is to great, more than a right angle, so much just is the other to little, so that both together are equal to two right angles, as you may perceive.    

The seventh Theorem. If two lines be drawn to any one prick in an other line, and those two lines do make with the first line, two right angles, other such as be equal to two right angles, and that toward one hand, than those two lines do make one straight line.  

Example.  
A.B. is a straight line,  diagram on which there doth light two other lines one from D, and the other from C, but considering that they meet in one prick E, and that the angles on one hand be equal to two right corners (as the last theorem doth declare) therefore may D.E. and E.C. be counted for one right line.    

The eight Theorem. when two lines do cut one an other cross ways they do make their match angles equal.  

Example.  
What match angles are, I  diagram have told you in the definitions of the terms. And here A, and B. are match corners in this example, as are also C. and D, so that the corner A, is equal to B, and the angle C, is equal to D.    

The ninth Theorem. when so ever in any triangle the line of one side is drawn forth in length, that utter angle is greater than any of the two inner corners, that join not with it.  

Example.  
The triangle A. D. C  diagram hath his ground line A. C. drawn forth in length unto B, so that the utter corner that it maketh at C, is greater than any of the two inner corners that lie against it, and join not with it, which are A. and D, for they both are lesser than a right angle, and be sharp angles, but C. is a blonte angle, and therefore greater than a right angle.    

The tenth Theorem. In every triangle any two corners, how so ever you take them, are less than ij. right corners.  

Example.  
In the first triangle E, which is a  diagram threlyke, and therefore hath all his angles sharp, take any two corners that you will, and you shall perceive that they be lesser than two right corners, for in every triangle that hath all sharp corners (as you see it to be  diagram in this example) every corner is less than a right corner. And therefore also every two corners must needs be less than two right corners. Furthermore in that other triangle marked with M, which hath two sharp corners and one right, any two of them also are less than two right angles. For though you take the right corner for one, yet the other which is a sharp corner, is less than a right corner. And so it is true in all kinds of triangles, as you may perceive more plainly by the xxij Theorem.    

The xi Theorem. In every triangle, the greatest side lieth against the greatest angle.  

Example.  
As in this triangle A. B. C,  diagram the greatest angle is C. And A. B. (which is the side that lieth against it) is the greatest and longest side. And contrary ways, as A. C. is the shortest side, so B. (which is the angle lying against it) is the smallest and sharpest angle, for this doth follow also, that as the longest side lieth against the greatest angle, so it that followeth    

The twelft Theorem. In every triangle the greatest angle lieth against the longest side.  
For these ij. theorems are one in truth.   

The thirteenth theorem. In every triangle any ij. sides together how so ever you take them, are longer than the third.  
For example you shall take this triangle A.B.C. which hath a veery  diagram blunt corner, and therefore one of his sides greater a good deal then any of the other, and yet the ij. lesser sides together are great than it. And if it be so in a blunt angeled triangle, it must needs be true in all other, for there is no other kind of triangles that hath the one side so great above the other sides, as they that have blunt corners.   

The fourteenth theorem. If there be drawn from the ends of any side of a triangle two lines meeting within the triangle, those two lines shall be less than the other two sides of the triangle, but yet the corner that they make, shall be greater than that corner of the triangle, which standeth over it.  

Example.  
 diagram A.B.C. is a triangle. on whose ground line A.B. there is drawn ij. lines, from the ij. ends of it, I say from A. and B, and they meet within the triangle in the point D, wherefore I say, that as those two lines A.D. and B. D, are lesser than A.C. and B. C, so the angle D. is greater than the angle C, which is the angle against it.    

The fifteenth Theorem. If a triangle have two sides equal to the two sides of an other triangle, but yet the angle that is contained between those sides, greater than the like angle in the other triangle, then is his ground line greater than the ground line of the other triangle.  
Example.  diagram  
A.B.C. is a triangle, whose sides A.C. and B. C, are equal to E.D. and D. F, the two sides of the triangle D. E. F, but because the angle in D, is greater than the angle C. which are the ij. angles contained between the equal lines) therefore must the  diagram ground line E. F. needs be greater then the ground line A. B, as you see plainly.   

The xvi. Theorem. If a triangle have two sides equal to the two sides of an other triangle, but yet hath a longer ground line than that other triangle, then is his angle that lieth between the equal sides, greater than the like corner in the other triangle.  

Example.  
This Theorem is nothing else, but the sentence of the last Theorem turned backward, and therefore needeth none other proof neither declaration, than the other example.    

The seventeenth Theorem. If two triangles be of such sort, that two angles of the one be equal to ij. angles of the other, and that one side of the one be equal to on side of the other, whether that side do adjoin to one of the equal corners, or else lie against one of them, then shall the other two sides of those triangles be equal together, and the third corner also shall be equal in those two triangles.  

Example.  
 diagram Because that A. B. C, the one triangle hath two corners A. and B, equal to D. E, that are two corners of the other triangle. D. E. F. and that they have one side in them both equal, that is A. B, which is equa●l to D. E, therefore shall both the other ij. sides be equal one to an other, as A C. and B. C. equal to D. F and E. F, and also the third angle in them both shallbe equal, that is, the angle C. shallbe equal to t the angle F.    

The eighteenth Theorem. when on two right lines there is drawn a third right line cross ways, and maketh two match corners of the one line equal to the like two match corners of the other line, then are those two lines gemmow lines, or parallels.  

Example.  
The two first lines are A. B. and C. D, the third line that crosseth them is E. F.  diagram And because that E. F. maketh ij. match angles with A. B, equal to two other like match angles on C. D, (that is to say E. G, equal to K. F, and M. N. equal also to H, L.) therefore are those ij. lines A. B. and C. D. gemow lines, understand here by like match corners, those that go one way as doth E. G, and K. F, lykeways N. M, and H. L, for as E. G. and H. L, other N. M. and K. F. go not one way, so be not they like match corners.    

The nyntenth Theorem. when on two right lines there is drawn a third right line cross ways, and maketh the ij. over corners toward one hand equal together, then are those two lines parallels. And in like manner if two inner corners toward one hand, be equal to ii right angles.  

Example.  
As the Theorem doth speak of two over angles, so must you understand also of two neither angles, for the judgement is like in both. Take for an example the figure of the last theorem, where A. B, and C. D, be called parallels also, because E. and K, (which are two over corners) are equal, and like ways L. and M. And so are in like manner the neither corners N. and H, and G. and F. Now to the second part of the theorem, those two lines A. B. and C. D, shall be called parallels, because the ij. inner corners. As for example those two that be toward the right hand (that is G. and L.) are equal (by the first part of this nyntenth theorem) therefore must G. and L. be equal to two right angles.    

The xx. Theorem. when a right line is drawn cross over two right gemow lines, it maketh two match corners of the one line, equal to two match corners of the other line, and also both over corners of one hand equal together, and both neither corners likeways, and more over two inner corners, and two utter corners also toward one hand, equal to two right angles.  

Example.  
Because A. B. and C. D, (in the last figure) are parallels, therefore the two match corners of the one line, as E. G. be equal unto the two match corners of the other line, that is K. F, and likeways M. N, equal to H. L. And also E. and K. both over corners of the left hand equal together, and so are M. and L, the two over corners on the right hand, in like manner N. and H, the two neither corners on the left hand, equal each to other, and G. and F. the two neither angles on the right hand equal together.  
¶ furthermore yet G. and L. the two inner angles on the right hand be equal to two right angles, and so are M. and F. the two utter angles on the same hand, in like manner shall you say of N. and K. the two inner corners on the left hand. and of E. and H. the two utter corners on the same hand. And thus you see the agreeable sentence of these three theorems to tend to this purpose, to declare by the angles how to judge parallels, and contrary ways how you may by parallels judge the proportion of the angles.    

The xxi. Theorem. what so ever lines be parallels to any other line, those same be parallels together.  

Example.  
A. B. is a gemow line, or a parellele  diagram unto C. D. and E. F, like ways is a parallel unto C. D. Wherefore it followeth, that A. B. must needs be a parallel unto E. F.    

The xxij theorem. In every triangle, when any side is drawn forth in length, the utter angle is equal to the ij. inner angles that lie against it. And all iij. inner angles of any triangle are equal to ij. right angles.  

Example.  
The triangle being  diagram A. D. E. and the side A. E. drawn forth unto B, there is made an utter corner, which is C, and this utter corner C, is equal to both the inner corners that lie against it, which are A. and D. And all three inner corners, that is to say, A. D. and E, are equal to two right corners, whereof it followeth, that all the three corners of any one triangle are equal to all the three corners of every other triangle. For what so ever things are equal to any one third thing, those same are equal togitther, by the first common sentence, so that because all the three angles of every triangle are equal to two right angles, and all right angles be equal together (by the fourth request) therefore must it needs follow, that all the three corners of every triangle (accounting them together) are equal to iij. corners of any other triangle, taken all together.    

The xxiii theorem. when any ij. right lines doth touch and couple two other right lines, which are equal in length and parallels, and if those two lines be drawn toward one hand, then are they also equal together, and parallels.  

Example.  
A. B. and C. D. are ij. right lines and parallels, and equal  diagram in length, and they are touched and joined together by ij. other lines A. C. and B. D, this being so and A. C. and B. D. being drawn toward one side (that is to say, both toward the left hand) therefore are A, C. and B. D. both equal and also parallels.    

The xxiiij theorem. In any likeiamme the two contrary sides are equal together, and so are each two contrary angles, and the bias line that is drawn in it, doth divide it into two equal portions.  

Example.  
 diagram Here are two likeiams joined together, the one is a long square A. B. E, and the other is a losengelike D. C. E. F. which ij. likeiams are proved equal together, because they have one ground line, that is, F. E, And are made between one pair of gemow lines, I mean A. D. and E. H. By this Theorem may you know the art of the right measuring of likeiams, as in my book of measuring I will more plainly declare.    

The xxvi. Theorem. All likeiams that have equal ground lines and are drawn between one pair of parallels, are equal together.  

Example.  
first you must mark the difference between this Theorem and the last, for the last Theorem presupposed to the divers likeiams one ground line common to them, but this theorem doth presuppose a divers ground line for every like iamme, only meaning them to be equal in length, though they be divers in numbbe. As for example. In the last figure there are two parallels, A. D. and E. H, and between them are drawn three likeiams, the first is, A. B. E. F, the second is E. C. D. F, and the third is C. G. H. D. The first and the second have one ground line, (that is E. F.) and therefore in so much as they are between one pair of parallels, they are equal according to the five and twenty Theorem, but the third likeiamme that is C. G. H. D. hath his ground line G. H, several from the other, but yet equal unto it. wherefore the third likeiam is equal to the other two first likeiams. And for a proof that G.H. being the ground line of the third likeiamme, is equal to E. F, which is the ground line to both the other likeiams, that may be thus declared, G.H. is equal to C. D, saying they are the contrary sides of one likeiamme (by the four and twenty theorem) and so are C.D. and E. F. by the same theorem. Therefore saying both those ground lines. E.F. and G. H, are equal to one third line (that is C.D.) they must needs be equal together by the first common sentence.    

The xxvii. Theorem. All triangles having one ground line; an standing between one pair of parallels, are equal together.  

Example.  
 diagram A.B. and C. F. are two gemowe lines, between which there be made two triangles, A. D. E. and D. E. B, so that D. E, is the common ground line to them both. wherefore it doth follow, that those two triangles A.D.E. and D.E.B. are equal each to other.    

The xxviij. Theorem. All triangles that have like long ground lines, and be made between one pair of gemow lines, are equal together.  

Example.  
Example of this Theorem you may see in the last figure, where as six triangles made between those two gemowe lines A. B. and C. F, the first triangle is A. C. D, the second is A. D.E, the third is A. D.B, the fourth is A. B. E, the fift is D. E.B, and the sixth is B. E.F, of which fix triangles, A. D.E. and D. E. B. are equal, because they have one common ground line. And so likewise A.B.E. and A.B. D, whose comen ground line is A. B, but A.C.D. is equal to B. E.F, being both between one couple of parallels, not because they have one ground line, but because they have their ground lines equal, for C.D. is equal to E. F, as you may declare thus. C. D, is equal to A.B. (by the four and twenty Theorem) for theiare two contrary sides of one lykeiamme. A. C.D.B, and E. F by the same theorem, is equal to A. B, for they are the two the contrary sides of the likeiamme, A. E.F.B, wherefore C.D. must needs be equal to E.F. like wise the triangle A. C.D, is equal to A. B.E, because they are made between one pair of parallels and have their groundlines like, I mean C. D. and A. B. Again A. D.E, is equal to each of them both, for his ground line D. E, is equal to A. B, in so much as they are the contrary sides of one likeiamme, that is the long square A. B. D. E. And thus may you prove the equalness of all the rest.    

The xxix. Theorem. Alequal triangles that are made on one ground line, and rise one way, must needs be between one pair of parallels.  

Example.  
Take for example A. D.E, and D. E.B, which as the xxvij. conclusion doth prove) are equal together, and as you see, they have on ground line D.E. And again they rise toward one side, that is to say, up ward toward the line A. B, wherefore they must needs be enclosed, between one pair of parallels, which are here in this example A.B. and D.E.    

The thirty Theorem. Equal triangles that have the irground lines equal, and be drawn toward one side, are made between one pair of parallels.  

Example.  
The example that declared the last theorem, may well serve to the declaration of this also. For those ij. theorems do differ but in this one point, that the last theorem meaneth of triangles, that have one ground line common to them both, and this theorem doth presuppose the ground lines to be divers, but yet of one length, as A. C. D, and B. E.F, as they are ij. equal triangles approved, by the eight and twenty Theorem, so in the same Theorem it is declared, that their ground lines are equal together, that is C. D, and E. F, now this being true, and considering that they are made toward one side, it followeth, that they are made between one pair of parallels when I say, drawn toward one side, I mean that the triangles must be drawn other both upward from one parallel, other else both downward, for if the one be drawn upward and the other downward, then are they drawn between two pair of parallels, presupposing one to be drawn by their ground line, and then do they rise toward contrary sides.    

The xxxi, theorem. If a likeiamme have one ground line with a triangle, and be drawn between one pair of parallels, then shall the likeiamme be double to the triangle.  

Example.  
A. H. and B.G. are two gemow  diagram lines, between which there is made a triangle B. C G, and a lykeiamme, A.B.G. C, which have a ground line, that is to say, B. G. Therefore doth it follow that the lykeiamme A.B.G.C. is double to the triangle B. C. G. For every half of that lykeiamme is equal to the triangle, I mean A.B.F.E. other F.E. C.G. as you may conjecture by the xi conclusion geometrical.  
And as this Theorem doth speak of a triangle and likeiamme that have one groundelyne, so is it true also, if their groundelynes be equal, though they be divers, so that theibe made between one pair of parallels. And hereof may you perceive the reason, why in measuring the plat of a triangle, you must multiply the perpendicular line by half the ground line, or else the hole ground line by half the perpendicular, for by any of these both ways is there made a lykeiamme equal to half such a one as should be made on the same hole ground line with the triangle, and between one pair of parallels. Therefore as that lykeiamme is double to the triangle, so the half of it, must needs be equal to the triangle. Compare the xu conclusion with this theorem.    

The xxxij Theorem. In all likeiams where there are more than one made about one bias line, the fill squares of every of them must needs be equal.  

Example.  
first before I declare the examples,  diagram it shallbe meet to show the true understandyngof this heorem. Therefore by the Bias line, 
Bias line.  I mean that line, which in any square figure doth run from corner to corner. And every square which is divided by that bias line into equal halves from corner to corner (that is to say, into two equal triangles) those be counted to stand about one bias line, and the other squares, which touch that bias line, with one of their corners only, those do I call Fill squares, according to the greek name, 
Fill squares. 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉  which is anapleromata, and called in latin supplementa, because that they make one general square, including and enclosing the other divers squares, as in this example H. C. E. N. is one square likeiamme, and L. M. G. C. is an other, which both are made about one bias line, that is N. M, than K. L. H. C. and C. E. F. G. are two full squares, for they do fill up the sides of the two first square sykeiammes, in such sort, that of all them four is made one great general square K. M.F.N.  
Now to the sentence of the theorem, I say, that the two fill squres. H. K. L. C. and C. E. F. G. are both equal together, (as it shall be declared in the book of proofs) because they are the fill squres of two likeiams made about one bias line, as the example showeth. Confer the twelfth conclusion with this the oreme.    

The xxxiij. Theorem. In all right anguled triangles, the square of that side which lieth against the right angle, is equal to the two squares of both the other sides  

Example.  
A.B.C. is a triangle, having  diagram a right angle in B. Wherefore it followeth, that the square of A. C, (which is the side that lieth against the right angle) shall be as much as the two squares of A. B. and B. C. which are the other two sides. ¶ By the square of any sign, you must understand a figure made just square, having all his iiij. sides equal to that line, whereof it is the square, so is A. C.F, the square of A.C. Lykewais A. B. D. is the square of A. B. And B. C. E. is the square of B.C. Now by the numbered of the divisions in each of these squares, may you perceive not only what the square of any line is called, but also that the theorem is true, and expressed plainly both by lines and numbered. For as you see, the greater square (that is A.C.F.) hath five divisions on each side, all equal together, and those in the whole square are twenty and five. Now in the left square, which is A.B.D. there are but three of those divisions in one side, and that yieldeth nine in the whole. So lykeways you see in the mean square A.C.E. in every side four parts, which in the whole amount unto sixteen. Now add together all the parts of the two lesser squares, that is to say, sixteen and nine, and you perceive that they make twenty and five, which is an equal numbered to the sum of the greater square.  
By this theorem you may understand a ready way to know the side of any right anguled triangle that is unknown, so that you know the length of any two sides of it. For by turning the two sides certain into their squares, and so adding them together, other subtracting the one from the other (according as in the use of these theorem es I have set forth) and then finding the root of the square that remaineth, which root (I mean the side of the square) is the just length of the unknown side, which is sought for. But this appertaineth to the third book, and therefore I will speak no more of it at this time.    

The xxxiiij. Theorem. If so be it, that in any triangle, the square of the one side be equal to the two squares of the other ij. sides, than must needs that corner be a right corner, which is contained between those two lesser sides.  

Example.  
As in the figure of the last Theorem, because A. C, made in square, is as much as the square of A. B, and also as the quare of B.C. joined both together, therefore the angle that is enclosed between those two lesser lines, A.B. and B. C. (that is to say) the angle B. which lieth against the line, A. C, must needs be a right angle. This theorem doth so depend of the truth of the last, that when you perceive the truth of the one, you can not justly doubt of the others truth, for they contain one sentence, contrary ways pronounced.    

The xxxv theorem. If there be set forth two right lines, and one of them parted into sundry parts, how many or few so ever they be, the square that is made of those ij. right lines proposed, is equal to all the squares, that are made of the undivided line, and every part of the divided line.  

Example.  
 diagram The ij. lines proposed are A. B. and C. D, and the line A.B.  diagram is divided into three parts by E. and F. Now saith this theorem,  diagram that the square that is made of those two whole lines A.B. and C. D, so that the line A.B. standeth for the length of the square, and the other line C.D. for the breadth of the same. That square (I say) will be equal to all the squares that be made, of the undiveded line (which is C.D.) and every portion of the divided line. And to declare that particularly, first I make an other line G. K, equal to the line C. D, and the line G.H. to be equal to the line A. B, and to be divided into iij. like parts, so that G. M. is equal to A. E, and M.N. equal to E. F, and then must N. H. needs remain equal to F.B. Then of those ij. lines G. K, undivided, and G.H. which is divided, I make a square, that is G. H.K.L, In which square if I draw cross lines from one side to the other, according to the divisions of the line G. H, then will it appear plain, that the theorem doth affirm. For the first square G. M.O.K, must needs be equal to the square of the line C. D, and the first portion of the divided line, which is A. E, for because their sides are equal. And so the second square that is M. N.P.O, shall be equal to the square of C. D, and the second part of A. B, that is E.F. Also the third square which is N. H.L.P, must of necessity be equal to the square of C. D, and F. B, because those lines be so coupled that every couple are equal in the several figures. And so shall you not only in this example, but in all other find it true, that if one line be divided into sundry parts, and an other line whole and undivided, matched with him in a square, that square which is made of these two whole lines, is as much just and equally, as all the several squares, which be made of the whole line undivided, and every part severally of the divided line.    

The xxxvi. Theorem. If a right line be parted into ij. parts, as chance may hap, the square that is made of that whole line, is equal to both the squares that are made of the same line, and the two parts of it severally.  

Example.  
The line proponed being A.B. and divided, as chance happeneth, in C. into ij. unequal parts,  diagram I say that the square made of the hole line A. B, ' is equal to the two squares made of the same line with the two parts of itself, as with A. C, and with C. B, for the square D, E.F.G. is equal to the two other partial squares of D.H. K G and H. E. F. K, but that the greater square is equal to the square of the whole line A. B, and the partial squares equal to the squares of the second parts of the same line joined with the whole line, your eye may judge without much declaration, so that I shall not need to make more exposition thereof, but that you may examine it, as you did in the last Theorem.    

The xxxvij Theorem. If a right line be divided by chance, as it may happen, the square: hat is made of the whole line, and one of the parts of it which so ever it be, shall be equal to that square that is made of the ij. parts joined together, and to another square made of that part, which was before joined with the whole line.  

Example.  
The line A.B. is divided  diagram in C. into two parts, though not equally, of which two parts for an example I take the first, that is A. C, and of it I make one side of a square, as for example D. G. accounting those two lines to be equal, the other side of the square is D. E, which is equal to the whole line A.B. Now may it appear, to your eye, that the great square made of the whole line A. B, and of one of his parts that is A. C, (which is equal with D, G.) is equal to two partial squares, whereof the one is made of the said greater portion A. C, in as much as not only D. G, being one of his sides, but also D. H. being the other side, are each of them equal to A. C. The second square is H. E.F.K, in which the one side H. E, is equal to C. B, being the lesser part of the line, A. B, and E.F. is equal to A.C. which is the greater part of the same line. So that those two squares D. H.K.G, and H, E, F, K, be both of them no more than the great square D. E, F, G, according to the words of the Theorem afore said.    

The xxxviij. Theorem. If a right line be divided by chance, into parts, the square that is made of that whole line, is equal to both the squares that are made of each part of the line, and moreover to two squares made of the one portion of the divided line joined with the other in square.  

Example.   diagram 
So is that greatest square, being made of the hole line A. B, equal to the ij. squares of each of his parts severally, and more by as much just as two long squares, made of the longer portion of the divided line joined in square with the shorter part of the same divided line, as the theorem would. And as here I have put an example of a line divided into two parts, so the theorem is true of all divided lines of what number so ever the parts be, four, five, or two. etc.  
This theorem hath great use, not only ingeometrie, but also in arithmetic, as hereafter I will declare in convenient place    

The xxxix theorem. If a right line be divided into two equal parts, and one of these two parts divided again into two other parts, as happeneth the long square that is made of the third or later part of that divided line, with the residue of the same line, and the square of the mydlemoste part, are both together equal to the square of half the first line.  

Example.  
The line A.B. is divided into  diagram ij. equal parts in C, and that part C.B. is divided again as happeneth in D. Wherefore saith the Theorem that the long square made of D. B. and A. D, with the square of C.D. (which is the L middle portion) shall both be equal to the square of half the line A. B, that is to say, to the square of A. C, or else of C. D, which make all one. The long square F.G.N. O. which is the long square that the theorem speaketh of, is made of two long squares, whereof the first is F. G. M.K, and the second is K.N. O. M. The square of the middle portion is L. M. O.P. And the square of the halft of the first line is E. K. Q. L. Now by the theorem, that long square F. G. M. O, with the just square L.M. O. P, must be equal to the great square E.K.Q. L, which thing because it seemeth somewhat difficult to understand, although I intend not here to make demonstrations of the Theorems, because it is appointed to be done in the new edition of Euclid, yet I will show you briefly how the equality of the parts doth stand. And first I say, that where the comparison of equality is made between the great square (which is made of half the line A. B.) and two other, where of the first is the long square F.G. N. O, and the second is the full square L. M. O.P, which is one portion of the great square all ready, and so is that long square K. N.M.O, being a parcel also of the long square F. G.O.O, Wherefore as those two parts are common to both parts compared in equality, and therfire ●eynge both abated from each part, if the rest of both the other parts be equal, than were those whole parts equal before: Now the resle of the great square, those two lesser squares being taken away) is that long square E. N. P. Q, which is equal to the long square F.G. K. M, being the rest of the other part. And that they two be equal, their sides do declare. For the longest lines that is F. K, and E.Q. are equal, and so are the shorter lines, F. G, and E. N, and so appeareth the truth of the Theorem.    

The xl theorem. If a right line be divided into two even parts, and an other right line annexed to one end of that line, so that it make one right line with the first. The long square that is made of this whole line so augmented, and the portion that is added, with the square of half the right line, shall be equal to the square of that line, which is compounded of half the first line, and the part newly added.  

Example.  
The first line proponed is A. B, and it is divided into  diagram ij. equal parts in C, and an other right line, I mean B. D, annexed to one end of the first line. Now say I, that the long square A. D. M. K, is made of the whole line so augmented) that is A. D, and the portion annexed, that is D.M, for D.M is equal to B. D, wherefore that long square A.D. M. K, with the square of half the first line, that is E. G.H.L, is equal to the great square E.F.D. C. which square is made of the line C. D. that is to say, of a line compounded of half the first line, being C. B, and the portion annexed, that is B. D. And it is easily perceived, if you consider that the long square A. C. L. K. (which only is left out of the great square) hath another long square equal to him, and to supply his steed in the great square, and that is G, F.M. H. For their sides be of like lines in length.    

The xli. Theorem. If a rightline be divided by chance, the square of the same whole line, and the square of one of his parts are just equal to the long square of the whole line, and the said part twice taken, and more over to the square of the other part of the said line.  

Example.  
A. B. is the line divided in C. And D. E.F.G, is the square of the whole  diagram line, D.H. K. M. is the square of the lesser portion (which I take for an example) and therefore must be twice reckoned. Now I say that those ij. squares are equal to two long squares of the whole line A. B, and his said portion A. C, and also to the square of the other portion of the said first line, which portion is C. B, and his square K. N. F. L. In this theorem there is no difficulty, if you consider that the little square D.H.K.M. is four times reckoned, that is to say, first of all as a part of the greatest square, which is D.E.F.G. Secondly he is rekned by himself. thirdly he is accounted as parcel of the long square D. E.N.M, And four he is taken as a part of the other long square D. H.L.G, so that in as much as he is twice reckoned in one part of the comparison of equality, and twice also in the second part, there can rise none occasion of error or doubtfulness thereby.    

The xlij. Theorem. If a right line be divided as chance happeneth the iiij. long squares, that may be made of that whole line and one of his parts with the square of the other part, shall be equal to the square that is made of the whole line and the said first portion joined to him in length as one whole line.  

Example.  
The first line is A. B, and is divided by C. into two unequal  diagram parts as happeneth. the longsquare of it and his lesser portion A. C, is four  diagram times drawn, the first is E. G.M. K, the second is K. M. Q.O, the third is H. K. R. S, and the fourth is K.L.S. T. And where as it appeareth that one of the little squares (I mean K. L.P O) is reckoned twice, once as par cell of the second longsquare and again as part of the third longsquare, to avoid ambiguity, you may place one instead of it, an other square of equality, with it. that is to say, D. E.K.H, which was at no time accompethy as parcel of any one of them, and then have you iiij. long squares distinctly made of the whole line A. B, and his lesser portion A.C. And within them is there a great full square P.Q. T. V. which is the just square of B. C, being the greater portion of the line A.B. And that those five squares do make just as much as the whole square of that longer line D. G, (which is as long as A. B, and A.C. joined together) it may be judged easily by the eye, sith that one great square doth comprehend in it all the other five squares, that is to say, four longsquares (as is before mentioned) and one full square. which is the intent of the Theorem.    

The xliij. Theorem. If a right line be divided into ij. equal parts first, and one of those parts again into other ij. parts, as chance happeneth, the square that is made of the last part of the line so divided, and the square of the residue of that whole line, are double to the square of half that line, and to the square of the middle portion of the same line.  

Example.  
The line to be divided is A. B, and is parted in C. into two equal parts, and then C. B, is divided again into two parts in D, so that the meaning of the Theorem, is that the square of D. B. which is the latter part of the line, and the  diagram square of A. D, which is the residue of the whole line. Those two squares, I say, are double to the square of the one half of the line, and to the square of C. D, which is the middle portion of those three divisions. Which thing that you may more easily perceive, I have drawn four squares, whereof the greatest being marked with F. is the square of A.D. The next, which is marked with G, is the square of half the line, that is, of A. C, And the other two little squares marked with F. and H, be both of one bigness, by reason that I did divide C.B. into two equal parts, so that you may take the square F, for the square of D. B, and the square H, for the square of C.D. Now I think you doubt not, but that the square E. and the square F, are double so much as the square G. and the square H, which thing the eayser is to be understand, because that the great square hath in his side iij. quarters of the first line, which multiplied by itself maketh nine quarters, and the square F. containeth but one quarter, so that both do make ten quarters. Then G. containeth iiij. quarters, saying his side containeth two, and H. containeth but one quarter, which both make but five quarters, and that is but half of ten. Whereby you may easily conjecture, that the meaning of the theorem is verified in the figures of this example.    

The xliiij. Theorem. If a right line be divided into ij. parts equally, and an other portion of a right line annexed to that first line, the square of this whole line so compounded, and the square of the portion that is annexed, are double as much as the square of the half of the first line, and the square of the other half joined in one with the annexed portion, as one whole line.  

Example.  
 diagram  
The line is A. B, and is divided first tnto two equal parts in C, and then is there annexed to it an other portion which is B.D. Now saith the Theorem, that the square of A. D, and the square of B. D, are double to the square of A. C, and to the square of C.D. The line A.B. containing four parts, then must needs his half contain ij. parts of such parts I suppose B.D. (which is the annexed line) to contain three, so shall the hole line comprehend seven. parts, and his square xlix. parts, whereunto if you ad the square of the annexed line, which maketh nine, than those both do yield, lviij. which must be double to the square of the half line with the annexed portion. The half line by itself containeth but two parts, and therefore his square doth make four. The half line with the annexed portion containeth five, and the square of it is. xxv, now put four to. xxv, and it maketh just. xxix, the even half of fifty and eight, whereby appeareth the truth of the theorem.    

The xlv theorem. In all triangles that have a blunt angle, the square of the side that lieth against the blunt angle, is greater than the two squares of the other two sides, by twice as much as is comprehended of the one of those two sides (enclosing the blunt corner) and that portion of the same line, being drawn forth in length, which lieth between the said blunt corner and a perpendicular line lighting on it, and drawn from one of the sharp angles of the foresaid triangle.  

Example.  
For the declaration of this theorem and the next also, whose use are wonderful in the practice of Geometry, and in measuring especially, it shall be needful to declare that every triangle that hath no right angle, as those be which are called (as in the book of practice is declared) sharp cornered triangles, and blunt covered triangles, yet may they be brought to have a right angle, either by parting them into two lesser triangles, or else by adding an other triangle unto them, which may be a great help for the aid of measuring, as more largely shall be set forth in the book of measuring. But for this present place, this form will I use, (which Theon also useth) to add one triangle unto an other, to bring the blunt cornered triangle into a right angled triangle, whereby the proportion of the squares of the sides in such a blunt cornered triangle may the better be known.   diagram 
The square of the line A. B, is the great square marked with E. The square of A. C, is the mean square marked with F. The square of B. C, is the least square marked with G. And the long square marked with K, is set in steed of two squares made of B. C, and C. D. For as the shorter side is the just length of C. D, so the other longer side is just twice so long as B. C, Wherefore I say now according to the Theorem, that the great square E, is more than the other two squares F. and G, by the quantity of the long square K, whereof I reserve the proof to a more convenient place, where I will also teach the reason how to find the length of all such perpendicular lines, and also of the line that is drawn between the blunt angle and the perpendicular line, with sundry other very pleasant conclusions.    

The xlvi Theorem. In sharp cornered triangles, the square of any side that lieth against a sharp corner, is lesser than the two squares of the other two sides, by as much as is comprised twice in the long square of that side, on which the perpendicular line falleth, and the portion of that same line, lying between the perpendicular, and the foresaid sharp corner.  

Example.  
first I set  diagram forth the triangle A. B. C, and in it I draw a plumb line from the angle C. unto the line A. B, and it lighteth in D. Now by the theorem the square of B.C. is not so much as the square of the other two sides, that of B. A. and of A.C. by as much as is twice contained in the long square made of A. B, and A. D, A. B. being the line or side on which the perpendicular line falleth, and A.D. being that portion of the same line which doth lie between the perpendicular line, and the said sharp angle limited, which angle is by A.  
For declaration of the figures, the square marked with E. is the square of B. C, which is the side that lieth against the sharp angle, the square marked with C. is the square of A. B, and the square marked with F. is the square of A. C, and the two long squares marked with H. K, are made of the hole line A. B, and one of his portions A. D. And truth it is that the square E. is lesser than the other two squares C. and F. by the quantity of those two long squares H. and K. Whereby you may consider again, an other proportion of equality, that is to say, that the square E. with the two long squares H. K, are just equal to the other two squares C. and F. And so may you make, as it were an other theorem. That in all sharp cornered triangles, where a perpendicular line is drawn from one angle to the side that lieth against it, the square of any one side, with the ij. longesquares made of that hole line, whereon the perpendicular line doth light, and of that portion of it, which joineth to that side, whose square is all ready taken, those three figures, I say, are equal to the ij. squares, of the other ij. sides of the triangle. In which you must understand, that the side on which the perpendicular falseth, is thrice used, yet is his square but ones mentioned, for twice he is taken for one side of the two long squares. And as I have thus made as it were an other theorem out of this forty and six theorem, so might I out of it, and the other that goeth next before, make as many as would suffice for a whole book, so that when they shall be applied to practise, and consequently to express their benefit, no man that hath not well weighed their wonderful commodity, would credit the possibility of their wonderful use, and large aid in knowledge. But all this will I remit to a place convenient.    

The xlvij Theorem. If ij. points be marked in the circumference of a circle, and a right line drawn from the one to the other, that line must needs fall with in the circle.  

Example.  
The circle is A. B.C.D, the ij. points are A. B, the right line that is drawenne from the one to the other, is the line A. B, which as you see, must needs light  diagram within the circle. So if you put the points to be A. D, or D. C, or A. C, other B. C, or B. D, inany of these cases you see, that the line that is drawn from the one prick to the other doth evermore run within the edge of the circle, else can it be no right line. Howbeit, that a crooked line, especially being more crooked than the portion of the circumference, may be drawn from point to point without the circle. But the theorem speaketh only of right lines, and not of crooked lines.    

The xlviij. Theorem. If a right line passing by the centre of a circle, do cross an other right line within the same circle, passing beside the centre, if be divide the said line into two equal parts, then do they make all their angles right. And contrary ways, if they make all their angles right, then doth the longer line cut the shorter in two parts.  

Example.  
The circle is A. B. C. D, the line that passeth by the centre, is A. E. C, the line that goeth beside the centre is D. B. Now  diagram say I, that the line A. E. C, doth cut that other line D. B. into two just parts, and therefore all their four angles are right angles. And contrary ways, because all their angles are right angles, therefore it must be true, that the greater cutteth the lesser into two equal parts, according as the Theorem would.    

The xlix. Theorem. If two right lines drawn in a circle do cross one an other, and do not pass by the centre, every of them doth not divide the other into two equal partions.  

Example.  
 diagram The circle is A. B. C. D, and the centre is E, the one line A. C, and the other is B. D, which two lines cross one an other, but yet they go not by the centre, wherefore according to the words of the theorem, each of them doth cut the other into equal portions. For as you may easily judge, A C. hath one portion longer and an other shorter, and so like wise B. D. Howbeit, it is not so to be understand, but one of them may be divided into ij. even parts, but both to be cut equally in the middle, is not possible, unless both pass through the centre, therefore much rather when both go beside the centre, it can not be that each of them should be justly parted into ij. even parts.    

The L. Theorem. If two circles cross and cut one an other, then have not they both one centre.  

Example.  
This theorem seemeth of itself  diagram so manifest, that it needeth neither demonstration neither declaration. Yet for the plain understanding of it, I have set forth a figure here, where ij. circles be drawn, so that one of them doth cross the other (as you see) in the points B. and G, and their centres appear at the first sight to be divers. For the centre of the one is F, and the centre of the other is E, which differ as far a sondre, as the edges of the circles, where they be most distant in sunder.    

The Li. Theorem. If two circles be so drawn, that one of them do touch the other, then have they not one centre.  

Example.  
There are two circles made, as you see, the one is A. B. C, and hath his centre by G, the other is  diagram B. D. E, and his centre is by F, so that it is easy enough to perceive that their centres do differ as much a sunder, as the half diameter of the greater circle is longer than the half diameter of the lesser circle. And so must it needs be thought and said of all other circles in like kind.    

The lij theorem. If a certain point be assigned in the diameter of a circle, distant from the centre of the said circle, and from that point diverse lines drawn to the edge and circumference of the same circle, the longest line is that which passeth by the centre, and the shortest is the residue of the same line. And of all the other lines that is ever the greatest, that is nighest to the line, which passeth by the centre. And contrary ways, that is shortest, that is farthest from it. And amongst them all there can be but only two equal together, and they must needs be so placed, that the shortest line shall be in the just middle betwixt them.  

Example.  
 diagram The circle is A. B. C. D. E. H, and his centre is F, the diameter is A. E, in which diameter I have taken a certain point distant from the centre, and that point is G, from which I have drawn four lines to the circumference, beside the two parts of the diameter, which maketh up vi. lines in all. Now for the diversity in quantity of these lines, I say according to the Theorem, that the line which goeth by the centre is the longest line, that is to say, A. G, and the reside we of the same diameter being G. E, is the shortest line. And of all the other that line is longest, that is nearest unto that part of the diameter which goeth by the centre, and that is shortest, that is farthest distant from it, wherefore I say, that G. B, is longèr than G. C, and therefore much more longer than G. D, sith G. C, also is longer than G. D, and by this may you soon perceive, that it is not possible to draw two lines on any one side of the diameter, which might be equal in length together, but on the one side of the diameter may you easily make one line equal to an other, on the other side of the same diameter, as you see in this example G. H, to be equal to G. B, between which the line G. E, (as the shortest in all the circle) doth stand even distant from each of them, and that is the precise knowledge of their equality, if they be equally distant from one half of the diameter. Where as contrary ways if the one be nearer to any one half of the diameter than the other is, it is not possible that they two may be equal in length, namely if they do end both in the circumference of the circle, and be both drawn from one point in the diameter, so that the said point be (as the Theorem doth suppose) somewhat distant from the centre of the said circle. For if they be drawn from the centre, then must they of necessity be all equal, how many so ever they be, is the definition of a circle doth import, without any regard how near so ever they be to the diameter, or how distant from it. And here is to be noted, that in this Theorem, by nearness and distance is understand the nearness and distance of the extreme parts of those lines where they touch the circumference. For at the other end they do all meet and touch.    

The liij Theorem. If a point be marked without a circle, and from it diverse lines drawn cross the circle, to the circumference on the other side, so that one of them pass by the centre, than that line which passeth by the centre shall be the longest of all them that cross the circle. And of tother lines those are longest, that be next unto it that passeth by the centre. And those are shortest, that be farthest distant from it. But among those parts of those lines, which end in the outward circumference, that is most shortest, which is part of the line that passeth by the centre, and amongst the other each, of them, the nearer they are unto it, the shorter they are, and the farther from it, the longer they be. And amongst them all there can not be more than two of any one length▪ and they two must be on the two contrary sides of the shortest line.  

Example.   diagram 
Take the circle to be A. B. C, and the point assigned without it to be D. Now say I, that if there be drawn sundry lines from D, and cross the circle, ending in the circumference on the contrary side, as here you see, D. A, D. E, D. F, and D. B, then of all these lines the longest must needs be D. A, which goeth by the centre of the circle, and the next unto it, that is D. E, is the longest amongst the rest. And contrary ways, D. B, is the shortest, because it is farthest distant from D.A. And so may you judge of D. F, because it is nearer unto D. A, then is D. B, therefore is it longer than D. B. And likeways because it is farther of from D. A, then is D. E, therefore is it shorter than D.E. Now for those parts of the lines which be without the circle (as you see) D. C, is the shortest, because it is the part of that line which passeth by the centre, And D. K, is next to it in distance, and therefore also in shortness, so D. G, is farthest from it in distance, and thenrfore is the longest of them. Now D. H, being nearer than D. G, is also shorter than it, and being farther of, then D. K, is longer than it▪ So that for this part of the theorem (as I think) you do plain lie perceive the truth thereof, so the residue hath no difficult. For seeing that the nearer any line is to D. C, (which joineth with the diameter) the shorter it is and the farther of from it, the longer it is. And seeing two lines can not be of like distance being both on one side, therefore if they shall be of one length, and consequently of one distance, they must needs be on contrary sides of the said line D. C. And so appeareth the meaning of the whole Theorem.  
And of this Theorem doth there follow we an other like, which you may call other a theorem by itself, or else a Corollary unto this last theorem, I pass not so much for the name. But his sentence is this: when so ever any lines be drawn from any point, without a circle, whether they cross the circle, or cande in the utter edge of his circumference, those two lines that be equally distant from the least line are equal together, and contrary ways, if they be equal together, they are also equally distant from that least line.  
For the declaration of this proposition, it shall not need to use any other example, then that which is brought for the explication of this last theorem, by which you may without any teaching casyly perceive both the meaning and also the truch of this proposition.    

The L iiij. Theorem. If a point be set forth in a circle, and from that point unto the circumference many lines drawn, of which more than two are equal together, then is that point the centre of that circle.  

Example.  
 diagram  
The circle is A. B.C, and with init I have set fourth for an example three pricks, which are D.E. and F, and from every one of them I have drawn (at the least) iiij. lines unto the circumference of the circle but from D, I have drawn more, yet may it appear readily unto your eye, that of all the lines which be drawn from E. and F, unto the circumference, there are but two equal, and more can not be, for G.E. nor E.H. hath none other equal to them, nor can not have any being drawn from the fame point E. No more can L. F, or F. K, have any line equal to either of them, being drawn from the same point F. And yet from either of those two points are there drawn two lines equal together, as A. E, is equal to E. B, and B. F, is equal to F. C, but there can no third line be drawn equal to either of these two couples, and that is by reason that they be drawn from a point distant from the centre of the circle. But from D although there be seven lines drawn, to the circumference, yet all be equal, because it is the centre of the circle. And therefore if you draw never so many more from it unto the circumference, all shall be equal, so that this is the privilege (as it were of the centre) and therefore no other point can have above two equal lines drawn from it unto the circumference. And from all poittes you may draw ij. equal lines to the circumference of the cirle, whether that point be within the circle or without it.    

The l v. Theorem. No circle can cut an other circle in more points than two.  

Example.  
 diagram  
The first circle is A. B.F.E, the second circle is B. C. D, F, and they cross one an other in B. and in E, and in no more points. Nother is it possible that they should, but other figures there be, which may cut a circle in four parts, as you see in this example. where I have set forth one tun form, and one eye form, and each of them cutteth every of their two circles into four parts. But as they be irregulare forms, that is to say, such forms as have no precise measure neither proportion in their draught, so can there scarcely be made any certain theorem of them. But circles are regulare forms, that is to say, such forms as have in their protracture a just and certain proportion, so that certain and determinate truths may be affirmed of them, sith they are uniform and unchangeable.    

The lvi. Theorem. If two circles be so drawn, that the one be within the other, and that they touch one an other: If a line be drawn by both their centres, and so forth in length, that line shall run to that point, where the circles do touch.  

Example.  
 diagram  
The one circle, which is the greatest and uttermost is A. B. C, the other circle that is the lesser, and is drawn within the first, is A. D. E. The centre of the greater circle is F, and the centre of the lesser circle is G, the point where they touch is A. And now you may see the truth of the theorem so plainly, that it needeth no farther declaration. For you may see, that drawing a line from F. to G, and so forth in length, until it come to the circumference, it will light in the very point A, where the circles touch one an other.    

The Lvij Theorem. If two circles be drawn so one without and other, that their edges do touch and a right line be drawenne from the centre of the oneto the centre of the other, that line shall pass by the place of their touching.  

Example.  
The first circle is A. B.E, and his centre is K, The second circle is D, B. C, and his centre is H, the point where they do touch is B. Now do you see that the line K. H, which is drawn  diagram from K, that is centre of the first circle, unto H, being centre of the second circle, doth pass (as it must needs by the point B,) which is the very point where they do to touch together.    

The lviij theorem. One circle can not touch an other in more points than one, whether they touch within or without.  

Example.  
 diagram  
For the declaration of this Theorem, I have drawn iiij. circles, the first is A. B. C, and his centre H. the second is A. D. G, and his centre F. the third is L. M, and his centre K. the four is D. G.L.M, and his centre E. Now as you perceive the second circle A. D.G, toucheth the first in the inner side, inso much as it is drawn within the other, and yet it toucheth him but in one point, that is to say in A, so like ways the third him, as you may see, but in one place. And now as for the four circle, it is drawn to declare, the diversity between touching and cutting, or crossing. For one circle may cross and cut a great many other circles, yet can be not cut any one in more places than two, as the five and fifty Theorem affirmeth.    

The lix Theorem. In every circle those lines are to be counted equal, which are in like distance from the centre, And contrary ways they are in like distance from the centre, which be equal.  

Example.  
 diagram In this figure you see first the circle drawn, which is A. B.C.D, and his centre is E. In this circle also there are drawn two lines equally distant from the centre, for the line A. B, and the line D. C, are just of one distance from the centre, which is E, and therefore are they of one length. Again they are of one length (as shall be proved in the book of profess) and therefore then distance from the centre is all one.    

The lx Theorem. In every circle the longest line is the diameter, and of all the other lines, they are still longest that be next unto the centre, and they be the shortest, that be farthest distant from it.  

Example.  
 diagram In this circle A. B.C.D, I have drawn first the diameter, which is A. D, which passeth (as it must) by the centre E, Then have I drawn ij. other lines as M. N, which is nearer the centre, and F. G, that is farther from the centre, The fourth line also on the other side of the diameter, that is B. C, is nearer to the centre than the line F. G, for it is of like distance as is the line M.N. Now say I, that A. D, being the diameter, is the longest of all those lines, and also of any other that may be drawn within that circle, And the other line M. N, is longer than F. G, because it is nearer to the centre of the circle then F. G. Also the line F. G, is shorter than the line B. C. for because it is farther from the centre than is the line B. C. And thus may you judge of all lines drawn in any circle, how to know the proportion of their length, by the proportion of their distance, and contrary ways, how to discern the proportion of their distance by their lengths, if you know the proportion of their length. And to speak of it by the way, it is a marvelous thing to consider; that a man may know an exact proportion between two things, and yet can not name nor attain the precise quantity of those two things, As for exaunple, If two squares be set forth, whereof the one containeth in it five square feet, and the other containeth five and forty foot, of like square feet, I am not able to tell, no nor yet any man living, what is the precise measure of the sides of any of those two squares, and yet I can prove by unfallible reason, that their sides be in a triple proportion, that is to say, that the side of the greater square (which containeth xlv foot) is three times so long just as the side of the lesser square, that includeth but five foot. But this seemeth to be spoken out of season in this place, therefore I will omit it now, reserving the exacter declaration thereof to a more convenient place and time, and will proceed with the residue of the Theorems appointed for this book.    

The lxi Theorem. If a right line be drawn at any end of a diameter in perpendicular form, and do make a right angle with the diameter, that right line shall light without the circle, and yet so jointly knit to it, that it is not possible to draw any other right line between that said line and the circumference of the circle And the angle that is made in the semicircle is greater than any sharp angle that may be made of right lines, but the other angle without, is lesser than any that can be made of right lines.  

Example.  
In this circle A. B.C, the diameter is A. C, the perpendicular line, which maketh a right angle with the diameter, is E. A, which line falleth without the circle, and yet joineth so exactly unto it, that it is not possible to draw an other right line between the circumference of the circle and it, which thing  diagram is so plainly seen of the eye, that it needeth no farther declaration. For every man will easily consent, that between the crooked line A. F, (which is a part of the circumference of the circle) and A. E (which is the said perpendicular line) there can none other line be drawn in that place where they make the angle. Now for the residue of the theorem. The angle D. A. B, which is made in the semicircle, is greater than any sharp angle that may be made of right lines, and yet is it a sharp angle also, in as much as it is lesser than a right angle, which is the angle E. A.D, and the residue of that right angle, which lieth without the circle, that is to say, E. A.B, is lesser than any sharp angle that can be made of right lines also. For as it was before rehearsed, there can no right line be drawn to the angle, between the circumference and the right line E.A. Then must it needs follow, that there can be made no lesser angle of right lines. And again, if there can be no lesser than the one, then doth it soon appear, that there can be no greater than the other, for they two do make the whole right angle, so that if any corner could be made greater than the one part, then should the residue be lesser than the other part, so that other both parts must be false, or else both granted to be true.    

The lxij. Theorem. If a right line do touch a circle, and an other right line drawn from the centre of tge circle to the point where they touch, that line which is drawenne from the centre, shall be a perpendicular line to the touch line.  

Example.  
 diagram The circle is A. B. C, and his centre is F. The touch line is D. E, and the point where they touch is C. Now by reason that a right line is drawn from the centre F. unto C, which is the point of the touch, therefore saith the theorem, that the said line F. C, must needs be a perpendicular line unto the touch line D.E.    

The lxiij. Theorem. If a right line do touch a circle, and an other right line be drawn from the point of their touching, so that it do make right corners with the touch line, then shall the centre of the circle be in that same line, so drawn.  

Example.  
The circle is A. B. C, and the centre of it is G. The touch line is D. C.E, and the point where it toucheth, is C. Now  diagram it appeareth manifest, that if a right line be drawn from the point where the touch line doth join with the circle, and that the said line do make right corners with the touch line, then must it needs go by the centre of the circle, and then consequently it must have the said centre in him. For if the said line should go beside the centre, as F. C. doth, then doth it not make right angles with the touch line, which in the theorem is supposed.    

The lxiiij. Theorem. If an angle be made on the centre of a circle, and an other angle made on the circumference of the same circle, and their ground line be one common portion of the circumference, then is the angle on the centre twice so great as the other angle on the circumference  

Example.  
 diagram The cirle is A. B. C. D, and his centre is E: the angle on the centre is C. E.D, and the angle on the circumference is C. A. D t their comen ground line, is C. F.D, Now say I that the angle C. E. D, which is one the centre, is twice so great as the angle C. A.D, which is on the circumference.    

The lxv. Theorem. Those angles which be made in one cantle of a circle, must needs be equal together.  

Example.  
Before I declare this theorem by an example, it shall be needful to declare, what is to be understand by the words in this theorem. For the sentence can not be known, unless the very meaning of the words be first understand. Therefore when it speaketh of angel's made in one cantle of a circle, it is this to be understand, that the angle must touch the circumference: and the lines that do enclose that angle, must be drawn to the extremities of that line, which maketh the cantle of the circle. So that if any angle do not touch the circumference, or if the lines that in close that angle, do not end in the extremities of the cord line, but end other in some other part of the said cord, or in the circumference, or that any one of them do so end, then is not that angle accounted to be drawn in the said cantle of the circle. And this promised, now will I come to the meaning of the theorem. I set forth a circle which is A. B. C. D, and his  diagram centre E, in this circle I draw a line D. C, whereby there are made two cantels, a more and a lesser. The lesser is D. F. C, and the geater is D.A.C.C. In this greater cantle I draw two angles, the first is D. A.C, and the second is D. B.C which two angles by reason they are made both in one cantle of a circle (that is the cantle D.A.B. C) therefore are they both equal 〈…〉 Now doth there appear an other triangle, whose angle lighteth on the centre of the circle, and that triangle is D. E. C, whose angle is double to the other angles, as is declared in the lxiiij. Theorem, which may stand well enough with this Theorem, for it is not made in this cantle of the circle, as the other are, by reason that his angle doth not light in the circumference of the circle, but on the centre of it.    

The lxvi theorem. Every figure of four sides, drawn in a circle, hath his two contrary angles equal unto two right angles.  

Example.  
 diagram The circle is A. B. C. D, and the figure of four sides in it, is made of the sides B. C, and C. D, and D. A, and A.B. Now if you take any two angles that be contrary, as the angle by A, and the angle by C, I say that those two be equal to two right angles. Also if you take the angle by B, and the angle by D, which two are also contrary, those two angles are like ways equal to two right angles. But if any man will take the angle by A, with the angle by B, or D, they can not be accounted contrary, no more is not the angle by C. esteemed contrary to the angle by B, or yet to the angle by D, for they only be accounted contrary angles, which have no one line common to them both. Such is the angle by A, in respect of the angle by C, for there both lines be distinct, where as the angle by A, and the angle by D, have one common line A. D, and therefore can not be accounted contrary angles, So the angle by D, and the angle by C, have D. C, as a common line, and therefore be not contrary angles. And this may you judge of the residue, by like reason.    

The lxvij. Theorem. Upon one right line there can not be made two cantles of circles, like and unequal, and drawent toward one part.  

Example.  
Cantles of circles be then called like, when the angles that are made in them be equal. But now for the Theorem, let the right line be A. E.C, on which  diagram I draw a cantle of a circle, which is A.B.C. Now saith the Theorem, that it is not possible to draw an other cantle of a circle, which shall be unequal unto this first cantle, that is to say, other greater or lesser than it, and yet be like it also, that is to say, that the angle in the one shall be equal to the angle in the other. For as in this example you see a lesser cantle drawn also, that is A. D.C, so if an angle were made in it, that angle would be greater than the angle made in the cantle A. B. C, and therefore ban not they be called like cantess, but and if any other cantle were made greater than the first, than would the angle it it be lesser than that in the first, and so neither a lesser nother a greater cantle can be made upon one line with an other, but it will be unlike to it also.    

The lxviij Theorem. Like cantelles of circles made on equal right lines, are equal together.  

Example.  
What is mentby like cantles you have heard before. and it is easy to understand, that such figures are called equal, that be of one bigness, so that the one is neither greater nother lesser than the other. And in this kind of comparison, they must so agree, that if the one be laid on the other, they shall exactly agree in all their bounds, so that neither shall exceed other.  
 diagram Now for the example of the Theorem, I have set forth divers varieties of cantles of circles, amongst which the first and second are made upon equal lines, and are also both equal and like. The third couple are joined in one, and be neither equal, neither like, but expressing an absurd deformity, which would follow if this Theorem were not true. And so in the fourth couple you may see, that because they are not equal cantles, therefore can not they be like cantles, for necessarily it goeth together, that all cantles of circles made upon equal right lines, if they be like, they must be equal also.    

The lxix. Theorem. In equal circles, such angles as be equal are made upon equal arch lines of the circumference, whether the angle light on the circumference, or on the centre.  

Example.  
first I have set for an example two equal circles, that  diagram is A. B. C. D, whose centre is K, and the second circle E. F. G. H, and his centre L, and in each of them is there made two angles, one on the circumference, and the other on the centre of each circle, and they be all made on two equal arch lines, that is B.C.D. the one, and F.G.H. the other. Now saith the Theorem, that if the angle B. A. D, be equal to the angle F. E. H, then are they made in equal circles, and on equal arch lines of their circumference. Also if the angle B. K.D, be equal to the angle F. L.H, then be they made on the centres of equal circles, and on equal arch lines, so that you must compare those angles together, which are made both on the centres, or both on the circumference, and may not confer those angles, whereof one is drawn on the circumference, and the other on the centre. For evermore the angle on the centre in such sort shall be double to the angle on the circumference, as is declared in the three score and four Theorem.    

The lxx Theorem. In equal circles, those angles which be made on equal arch lines, are ever equal together, whether they be made on the centre, or on the circumference.  

Example.  
This Theorem doth but convert the sentence of the last Theorem before, and therefore is to be understand by the same examples, for as that saith, that equal angles occupy equal archesynes, so this saith, that equal arch lines causeth equal angles, considering all other circumstances, as was taught in the last theorem before, so that this theorem doth affirming speak of the equality of those angles, of which the last theorem spoke conditionally. And where the last theorem spoke affirmatively of the arch lines, this theorem speaketh conditionally of them, as thus: If the arch line B. C. D. be equal to the other arch line F. G.H, then is that angle B.A.D. equal to the other angle F.E.H. Or else thus may you declare it causally: Because the arch line B.C. D, is equal to the other arch line F. G.H, therefore is the angle B. K. D. equal to the angle F. L.H, considering that they are made on the centres of equal circles. And so of the other angles, because those two arch lines aforesaid are equal, therefore the angle D. A.B, is equal to the angle F. E. H, for as much as they are made on those equal arch lines, and also on the circumference of equal circles And thus these theorems do one declare an other, and one verify the other.    

The lxxi. Theorem. In equal circles, equal right lines being drawn, do cut away equal arch lines from their circumference, so that the greater arch line of the one is equal to the greater arch line of the other, and the lesser to the lesser.  

Example.  
The circle A.  diagram B. C. D, is made equal to the circle E. F.G.H, and the right line B. D. is equal to the right line F. H, wherefore it followeth, that the ij. arch lines of the circle A. B. D, which are cut from his circumference by the right line B. D, are equal to two other arch lines of the circle E. F. H, being cut from his circumference, by the right line F. H. that is to say, that the arch line B. A. D, being the greater arch line of the first circle, is equal to the arch line F. E. H, being the greater arch line of the other circle. And so in like manner the lesser arch line of the first circle, being B. C.D, is equal to the lesser arch line of the second circle, that is F.G.H.    

The lxxij. Theorem. In equal circles, under equal arch lines the right lines that be drawn are equal together.  

Example.  
This Theorem is none other, but the conversion of the last Theorem before, and therefore needeth none other example. For as that did declare the equality of the arch lines, by the equality of the right lines, so do the this Theorem declare the equalness of the right lines to ensue of the equalness of the arch lines, and therefore declareth that right line B. D, to be equal to the other right line F. H, because they both are drawn under equal arch lines, that is to say, the one under B. A.D, and tother under F. E.H, and those two arch lines are esteemed equal by the theorem last before, and shall be proved in the book of proofs.    

The lxxiij. Theorem. In every circle, the angle that is made in the half circle, is a just right angle, and the angle that is made in a cantle greater than the half circle, is lesser than a right angle, but that angle that is made in a cantle, lesser than the half circle, is greater than a right angle. And moreover the angle of the greater cantle is greater than a right angle and the angle of the lesser cantle is lesser than a right angle.  

Example.  
In this proposition, it shall be meet to note, that there is a great diversity between an angle of a cantle, and an angle made in a cantle, and also between the angle of a semicircle, and the angle made in a semicircle. Also it is meet to note that all angles that be made in the part of a circle, are made other in a semicircle (which is the just half circle) or else in a cantle of the circle, which cantle is other greater or lesser than the semicircle is, as in this figure annexed you may perceive every one of the things severally.  
 diagram first the circle is, as you see, A. B.C.D, and his centre E, his diameter is A. D, Then is there a line drawn from A. to B, and so forth unto F, which is without the circle: and an other line also from B. to D, which maketh two cantles of the whole circle, The greater cantle is D. A B, and the lesser cantle is B. C. D, In which lesser cantle also there are two lines that make an angle, the one line is B. C, and the other line is C.D. Now to show the difference of an angle in a cantle, and an angle of a cantle, first for an example I take the greater cantle B. A.D, in which is but one angle made, and that is the angle by A, which is made of the line A, B, and the line A. D, And this angle is therefore called an angle in a cantle. But now the same cantle hath two other angles, which be called the angles of that cantle, so the two angles made of the right line D. B, and the arch line D. ●. B, are the two angles of this cantle, whereof the one is by D, and the other is by B. Where you must remember, that the angle by D. is made of the right line B. D, and the arch line D.A. And this angle is divided by an other right line A. E.D, which in this case must be omitted as no line. Also the angle by B. is made of the right line D. B, and of the arch line. B. A, & although it be divided with ij. other right lines, of which the one is the right line B. A, & tother the right line B. E, yet in this case they are not to be considered. And by this may you perceive also which be the angles of the lesser cantle, the first of them is made of that right line B. D, & of the arch line B. C, the second is made of the right line. D. B, & of the arch line D.C. Then are there ij. other lines, which divide those ij. corners, that is the line B. C, & the line C. D, which ij. lines do meet in the point C, and there make an angle, which is called an angle made in that lesser cantle, but yet is not any angle of that cantle. And so have you heard the difference between an angle in a cantle, and an angle of a cantle. And ●n like sort shall you judge of the angle made in a semicircle, which is distinct from the angles of the semicircle. For in this figure, the angles of the semicircle are those angles which be by A. and D, and be made of the right line A. D, being the diameter, and of the half circumference of the circle, but the angle made in the semicircle is that angle by B, which is made of the right line A. B, and that other right line B. D, which as they meet in the circumference, and make an angle, so they end with their other extremities at the ends of the diameter. These things premised, now say I touching the Theorem, that every angle that is made in a semicircle, is a right angle, and if it be made in any cantle of a circle, them must it neds be other a blunt angle, or else a sharp angle, and in no wise a right angle. For if the cantle wherein the angle is made, be greater than the half circle, then is that angle a sharp angle. And generally the greater the cantle is, the lesser is the angle comprised in that cantle: and contrary ways, the lesser any cantle is, the greater is the angle that is made in it. Wherefore it must needs follow, that the angle made in a cantle less than a semicircle, must needs be greater than a right angle. So the angle by B, being made of the right line A. B, and the right line B. D, is a just right angle, because it is made in a semicircle. But the angle made by A, which is made of the right line A. B, and of the right line A. D, is lesser than a right angle, and is named a sharp angle, for as much as it is made in a cantle of a circle, greater than a semicircle. And contrary ways, the angle by C, being made of the right line B. C, and of the right line C. D, is greater than a right angle, and is named a blunt angle, because it is made in a cantle of a circle, lesser than a semicircle. But now touching the other angles of the cantles, I say according to the Theorem, that the two angles of the greater cantle, which are by B. and D, as is before declared, are greater each of them then a right angle. And the angles of the lesser cantle, which are by the same letters B, and D, but be on the other side of the cord, are lesser each of them then a right angle, and be therefore sharp corners.    

The lxxiiij. Theorem. If a right line do touch a circle, and from the point where they touch, a right line be drawn cross the circle, and divide it, the angles that the said line doth make with the touch line, are equal to the angles which are made in the cantles of the same circle, on the contrary sides of the line aforesaid.  

Example.  
The circle is A. B.C.D, and  diagram the touch line is E. F. The point of the touching is D, from which point I suppose the line D. B, to be drawn cross the circle, and to divide it into two cantles, whereof the greater is B. A.D and the lesser is B. C.D, and in each of them an angle drawn, for in the greater cantle the angle is by A, and is made of the right lines B. A, and A. D, in the lesser cantle the angle is by C, and is made of the right lines B. C, and C.D. Now saith the Theorem that the angle B. D. F, is equal to the angle made in the cantle on the other side of the said line, that is to say, in the cantle B. A.D, so that the angle B. D.F, is equal to the angle B. A.D, because the angle B. D.F, is on the one side of the line B. D, (which is according to the supposition of the Theorem drawn cross the circle) and the angle B. A.D, is in the cantle on the other side. likeways the angle B. D.E, being on the one side of the line B. D, must be equal to the angle B. C.D, (that is the angle by C,) which is made in the cantle on the other side of the right line B.D. The proof of all these I do reserve, as I have often said, to a convenient book, wherein they shall be all set at large.    

The .lxxv. Theorem. In any circle when two right lines do cross one an other, the likeiamme that is made of the portions of the one line, shall be equal to the lykeiamme made of the parts of the other line.  
 diagram Because this Theorem doth serve to many uses, and would be well understand, I have set forth two examples of it. In the first, the lines by their crossing do make their portions somewhat toward an equality In the second the portions of the lines be very far from an equality, and yet in both these and in all other the Theorem is true. In the first example the circle is A. B.C.D, in which th'one line A. C, doth cross tother line B. D, in the point E. Now if you do make one likeiamme or longsquare of D. E, & E. B, being the two portions of the line D. B, that longsquare shall be equal to the other longsquare made of A. E, and E. C, being the portions of the other line A.C. Lykewaies in the second example, the circle is F. G.H.K, in which the line F. H, doth cross the other line G. K, in the point L. Wherefore if you make a lykeiamme or longsquare of the two parts of the line F. H, that is to say, of F. L, and L. H, that longsquare will be equal to an other longsquare made of the two parts of the line G. K. which parts are G. L, and L.K. Those longsquares have I set forth under the circles containing their sides, that you may somewhat whet your own wit in practising this Theorem, according to the doctrine of the nineteenth conclusion.   

The lxxvi Theorem. If a point be marked without a circle, and from that point two right lines drawn to the circle, so that the one of them do run cross the circle, and the other do touch the circle only, the long square that is made of that whole line which crosseth the circle, and the portion of it, that lieth between the utter circumference of the circle and the point, shall be equal to the full square of the other line, that only toucheth the circle.  

Example.  
The circle is D. B.C, and the point without the circle is A, from which point there is drawn one line cross the circle, and that is A. D.C, and an other line is drawn from the said prick to the marge or edge of the circumference of the circle, and doth only touch it, that is the line A.B. And of that first line A. D.C, you may perceive one part of it, which is A. D,  diagram to lie without the circle, between the utter circumference of it, and the point assigned, which was A. No we concerning the meaning of the Theorem, if you make a longsquare of the whole line A. C, and of that part of it that lieth between the circumference and the point, (which is A. D,) that long square shall be equal to the full square of the touch line A. B, according not only as this figure showeth, but also the said nineteenth conclusion doth prove, if you list to examine the one by the other.    

The lxxvij Theorem. If a point be assigned without a circle, and from that point two right lines be drawn to the circle, so that the one do cross the circle, and the other do end at the circumference, and that the longsquare of the line which crosseth the circle made with the portion of the same line being without the circle between the utter circumference and the point assigned, do equally agree with the just square of that line that endeth at the circumference, then is that line so ending on the circumference a touch line unto that circle.  

Example.  
In as much as this Theorem is nothing else but the sentence of the last Theorem before converted, therefore it shall not be needful to use any other example than the same, for as in that other Theorem because the one line is a touch line, therefore it maketh a square just equal with the longsquare made of that whole line, which crosseth the circle, and his portion lying without the same circle. So saith this Theorem: that if the just square of the line that endeth on the circumference, be equal to that longsquare which is made as for his longer sides of the whole line, which cometh from the point assigned, and crosseth the circle, and for his other shorter sides is made of the portion of the same line, lying between the circumference of the circle and the point assigned, then is that line which endeth on the circumference a right touch line, that is to say, if the full square of the right line A. B, be equal to the longsquare made of the whole line A. C, as one of his lines, and of his portion A. D, as his other line, then must it needs be, that the line A. B, is a right touch line unto the circle D. B.C. And thus for this time I make an end of the Theorems.     
FINIS.   

IMPRINTED at London in Paul's churchyard, at the sign of the Brazen serpent, by Reynold wolf.   

Cum privilegio ad imprimendum solum.  
ANNO DOMINI. M.D.LI