THE ELEMENTS OF EUCLID Explained and Demonstrated in a New and most easie Method With the USES of each PROPOSITION In all the Parts of the MATHEMATICKS By Claude Francois Milliet D'Chales a Jesuite Done out of FRENCH Corrected and Augmented and Illustrated with Nine Copper Plates and the Effigies of EVCLID By Reeve Williams Philomath LONDON Printed for Philip Lea Globemaker at the Atlas and Hercules in the Poultrey near Cheapside 1685. To the Honourable Samuel Pepys Esq Principal Officer OF THE NAVY SECRETARY TO THE ADMIRALTY AND PRESIDENT OF THE ROYAL SOCIETY Honoured Sir THE Countenance and Encouragment you have always given to Mathematical Learning especially as it hath a tendency to Promote the Publique Good has emboldened me humbly to Present your Honour with this little Peice which hath the Admirable Euclid for its Author and the Learned D'Chales for the Commentator the excellency of the Subject with the Apt and Profitable Application thereof in its Vses did first induce me to Translate it for my own use the benefit and quickening in those Mathematical Studies that some professed to have received did prevail with me to make it Publique and the great Obligations I lie under from the many undeserved Favours of your Honour toward me I thought did engage me on this occasion to make some Publique Testimony and Acknowledgment thereof I therefore Humbly beg your Honours Patronage of this little Book and your Pardon for this Address intreating you will be pleased to look upon it with that Benign Aspect as you have been pleased always to vouchsafe to him who is Your Humble and most Obliged Servant REEVE WILLIAMS THE Authors Preface TO THE READER HAving long since observed that the greatest part of those that learn Euclid's Elements are very often dissatisfied therewith because they know not the use of Propositions so inconsiderable in appearance and yet so difficult I thought it might be to good purpose not only to make them as easie as possible but also to add some Vses after each Proposition to shew how they are applicable to Practice And this hath obliged me to change some of the Demnostrations which I looked upon to be too troublesome and above the usual reach of beginners and to substitute others more intelligible For this cause I have Demonstrated the Fifth Book after a method much more clear than that that makes use of equimultiples I have not given all the Vses of the Propositions for that would have made it necessary to recite all the Mathematicks and would have made the Book too big and too hard Wherefore I have only made choice of some of the plainest and easiest to conceive I would not have you to stand too much upon them nor make it your study to understand them perfectly because they depend upon other principles besides for which Reason I have distinguished them with the Italick Letter This is the design of this small Treatise which I willingly publish in a time when the Mathematicks are more than ever studied Milliet D'chales THE TRANSLATORS PREFACE THE Reader having perused the Authors Preface with this farther intimation that he will find the Subject and Scope of this work Succinctly and pertinently presented to him in the Argument before each particular book may I presume expect the loss from me and I shall not at all endeavor to bespeak the Readers acceptance of Euclid's Elements or persuade him to beleive the Necessary and Excellent Vsefulness thereof because every mans Experience so far as he understands them is an abundant Testimony thereunto Neither shall I need to commend the Method with the uses of our Author D'Chales who is well known to the learned of this Age by his Several excellent Mathematical Tracts * Cursus Mathe his Navigation his Local Motion c. for whosoever shall be a while conversant with this book may I presume fiind that instruction and incourgment in the learning of Euclid's Eliments as he hath not before met with in our English Tongue And this as it hath been my own experience since I Translated it from the French for the use of my English Schollars so it is one great cause of its coming abroad into the World for such as had Learned by it found it difficult to attain unless Transcribed which they thought tedious being a subject so Voluminous in Manuscript and full of Schemes I did therefore at their request and the importunity of some Friends condescend to the Printing hereof though not without much averseness to my own mind being unwilling to Expose my self in any Publick thing in this nice Critical Age But that difficulty being now overcome I shall only give the Reader to understand that I have faithfully rendred this Piece into English according to the sense of the Author but here and there omitting some small matters which I judged not so properly related to the subject of this Work and therein will make no want to the Reader nor I hope be any offence to the Ingenious Author himself I have only one thing more to add and that upon the account of an Objection I have met with that here is not all the Books of Euclid and it is true here is not all here are only the first six Books and the Eleventh and Twelfth the other being purposely omitted by our learned Author who judged the understanding of these to be sufficient for all the parts of the Mathematicks * See Argument before the Eleventh Book Page 304. and I could also give Instances of other excellent Authors that are of his Opinion and have taken the like course nay the truth is some very learned in the Mathematicks have reduced the Propositions of these Books to a much lesser Number and yet have thought them a compleat foundation to all the Sciences Mathematical but I shall not trouble my Reader farther on this account not doubting but when he hath perused and well considered our Euclid he will have a better Opinion thereof then any thing I can now say may Justly hope to beget in him and so I shall submit my whole Concernment herein to the Impartial Reader and remain ready to Serve him Reeve Williams Form my School at the Virginia Coff-house in St. Michaels Alley in Cornhill ADVERTISEMENT VVHereas in the French all the Definitions which needed all the Propositions as also all those uses our Author thought fit to Illustrate by Schemes were done in the Book in wood here they are in Copper plates to be placed at the end of the Book The Definitions and uses are in the Plates marked with Arithmetical Charactars or Ziphers but the Propsitions in Alphabetical Ziphers at the begining of the Book or at least when you come to the first Definition you are refâ—rred to the number of the Plate in Which you shall find the Scheme proper thereto as also the Def Prop. and uses belonging unto the Book unless the Plate could not contain them and then you are referred to the next Plate
whereon the head thereof you shall find the Book it belongeth to and the Propositions or uses continued in their order EIGHT BOOKS OF Euclid's Elements With the Vses of each PROPOSITION The FIRST BOOK EUCLID's Design in this Book is to give the first Principles of Geometry and to do the same Methodically he begins with the Definitions and Explication of the most ordinary Terms then he exhibits certain Suppositions and having proposed those Maxims which natural Reason teacheth he pretends to put forward nothing without Demonstration and to convince any one which will consent to nothing but what he shall be obliged to acknowledge in his first Propositions he treateth of Lines and of the several Angles made by their intersecting each other and having occasion to Demonstrate their Proprieties and compare certain Triangles he doth the same in the First Eight Propositions then teacheth the Practical way of dividing an Angle and a Line into two equal parts and to draw a Perpendicular he pursues to the propriety of a Triangle and having shewn those of Parallel Lines he makes an end of the Explication of this First Figure and passeth forwards to Parallelograms giving a way to reduce all sorts of Polygons into a more Regular Figure He endeth this Book with that Celebrated Proposition of Pythagoras and Demonstrates that in a Rectangular Triangle the Square of the Base is equal to the sum of the Squares of the Sides including the Right Angle DEFINITIONS 1. A Point is that which hath no part This Definition is to be understood in this sence The quantity which we conceive without distinguishing its parts or without thinking that it hath any is a Mathematical Point far differing from those of Zeno which were alltogether indivisible since one may doubt with a great deal of Reason if those last be possible which yet we cannot of the first if we conceive them as we ought 2. A Line is a length without breadth The sense of this Definition is the same with that of the foregoing the quantity which we consider having length without making any reflection on its breadth is that we understand by the word Line although one cannot draw a real Line which hath not a determinate breadth it is generally said that a Line is produced by the motion of a Point which we ought well to take notice of seeing that by a motion after that manner may be produced all sorts of quantities imagine then that a Point moveth and that it leaveth a trace in the middle of the way which it passeth the Trace is a Line 3. The Two ends of a Line are Points 4. A streight Line is that whose Points are placed exactly in the midst or if you would rather have it a streight Line is the shortest of all the Lines which may be drawn from one Point to another 5. A Superficies is a quantity to which is given length and breadth without considering the thickness 6. The extremities of a Superficies are Lines 7. A plain or straight Superficies is that whose Lines are placed equally between the extremities or that to which a streight Line may be applyed any manner of way Plate I. Fig. 1. I have already taken notice that motion is capable of producing all sorts of quantity whence we say that when a Line passeth over another it produces a superficies or a Plain and that that motion hath a likeness to Arithmetical Multiplication imagine that the Line AB moveth along the Line BC keeping the same situation without inclining one way or the other the Point A shall describe the Line AD the Point B the Line BC and the other Points between other Parallel Lines which shall compose the Superficies ABCD. I add that this motion corresponds with Arithmetical Multiplication for if I know the number of Points which are in the Lines AB BC Multiplying of them one by the other I shall have the number of Points which Composeth the Superficies ABCD as if AB contains four points and BC six saying Four times Six are Twenty Four the Superficies AB CD should be Composed of Twenty Four Points Now I may take for a Mathematical Point any quantity whatsoever for Example a Foot provided I do not subdivide the same into Parts 8. A plain Angle is the opening of Two Lines which intersect each other and which Compose not one single Line Fig. 2. As the opening D of the Lines AB CB which are not parts of the same Line A Right Lined Angle is the opening of two streight Lines It is principally of this sort of Angles which I intend to treat of at present because experience doth make me perceive that the most part of those who begin do mistake the measuring the quantity of an Angle by the length of the Lines which Composeth the same Fig. 3 4. The most open Angle is the greatest that is to say when the Lines including an Angle are farther asunder than those of another Angle taking them at the same distance from the Points of intersection of their Lines the first is greater than the Second so the Angle A is greater than E because if we take the Points B and D as far distant from the Point A as the Points G and L are from the Points E the Points B and D are farther asunder than the Points G and L from whence I conclude that if EG EL were continued the Angle E would be of the same Measure and less than the Angle A. We make use of Three Letters to express an Angle and the Second Letter denotes the Angular Point as the Angle BAD is the Angle which the Lines BA AD doth form at the Point A the Angle BAC is that which is formed by the Lines BA AC the Angle CAD is comprehended under the Line CA AD. Fig. 3. The Arch of a Circle is the measure of an Angle thus designing to measure the quantity of the Angle BAD I put one Foot of the Compasses on the Point A and with the other I describe an Arch of a Circle BCD the Angle shall be the greater by how much the Arch BCD which is the measure thereof shall contain a greater portion of a Circle and because that commonly an Arch of a Circle is divided into Three Hundred and Sixty equal Parts called Degrees It is said that an Angle containeth Twenty Thirty Forty Degrees when the Arch included betwixt its Lines contains Twenty Thirty Forty Degrees so the Angle is greatest which containeth the greatest number of Degrees As the Angle BAD is greater than GEL the Line CA divideth the Angle BAD in the middle because the Arches BC CD are equal and the Angle BAC is a part of BAD because the Arch BC is part of the Arch BD. 10. When a Line falling on another Line maketh the Angle on each side thereof equal Those Angles are Right Angles and the Line so falling is a Perpendicular Fig. 5. As if the Line AB falling on CD
the double Area of the Triangle I make use thereof in several other Propositions as in the seventh PROPOSITION XIV PROBLEM TO describe a Square equal to a right lined Figure given To make a Square equal to a Right lined Figure A make by the 45th of the 1st a Rectangle BC DE equal to the Right lined Figure A if its sides CD DC were equal we should have already our desire if they be unequal continue the Line BC until CF be equal to CD and dividing the Line BF in the middle in the Point G describe the Semicircle FHB then continue DC to H the Square of the Line CH is equal to the Right lined Figure A draw the Line GH Demonstration The Line BF is equally divided in G and unequally in C thence by the 5th the Rectangle comprehended under BC CF or CD that is to say the Rectangle BD with the Square CG is equal to the Square of GB or to its Equal GH Now by the 47th of the 1st the Square of GH is equal to the Square of CH CG therefore the Rectangle BD and the Square of CG is equal to the Squares of CG and of CH and taking away the Square of CG which is common to both the Rectangle BD or the Right lined Figure A is equal to the Square of CH. USE THis Proposition serveth in the first place to reduce into a Square any Right lined Figure whatever and whereas a Square is the first Measure of all Superficies because its Length and Breadth is equal we measure by this means all right lined Figures In the second place this Proposition teacheth us to find a mean Proportion between two given Lines as we shall see in the Thirteenth Proposition of the Sixth Book This Proposition may also serve to square curve lined Figures and even Circles themselves for any crooked or curve lined Figure may to sence be reduced to a Right lined Figure as if we inscribe in a Circle a Polygon having a thousand sides it shall not be sensibly different from a Circle and reducing the Polygone into a Square we square nearly the Circle THE THIRD BOOK OF Euclid's Elements THis Third Book explaineth the Propriety of a Circle and compareth the divers Lines which may be drawn within and without its Circumference It farther considereth the Circumstances of Circles which cut each other or which touch a streight Line and the different Angles which are made as well those in their Centers as in their Circumferences In fine it giveth the first Principles for establishing the Practice of Geometry by the which we make use and that very commodiously of a Circle in almost all Treatises in the Mathematicks DEFINITIONS 1. Def. of the 3 Book Those Circles are equal whose Diameters or Semidiameters are equal 2. Fig. 1. A Line toucheth a Circle when meeting with the Circumference thereof it cutteth not the same as the Line AB 3. Fig. 2. Circles touch each other when Meeting they cut not each other as the Circles AB and C. 4. Fig. 3. Right Lines in a Circle are equally distant from the Center when Perpendiculars drawn from the Center to those Lines are equal As if the Lines EF EG being Perpendiculars to the Lines AB CD are equal AB CD shall be equally distant from the Center because the Distance ought always to be taken or measured by Perpendicular Lines 5. Fig. 4. A Segment of a Circle is a Figure terminated on the one side by a streight Line and on the other by the Circumference of a Circle as LON LMN 6. The Angle of a Segment is an Angle which the Circumference maketh with a streight Line as the Angle OLN LMN 7. Fig. 5. An Angle is said to be in a Segment of a Circle when the Lines which form the same are therein as the Angle FGH is in the Segment FGH 8. Fig 6. An Angle is upon that Arch to which it is opposite or to which it serveth for a Base as the Angle FGH is upon the Arch FIH which may be said to be its Base 9. Fig. 6. A Sector is a Figure comprehended under two Semidiameters and under the Arch which serveth to them for a Base as the Figure FIGH PROPOSITION I. PROBLEM To find the Center of a Circle IF you would find the Center of the Circle AEBD draw the Line AB and divide the same in the middle in the Point C at which Point erect a Perpendicular ED which you shall divide also equally in the Point F. This Point F shall be the Center of the Circle for if it be not imagine if you please that the Point G is the Center draw the Lines GA GB GC Demonstration If the Point G were the Center the Triangles GAC GBC would have the sides GA GB equal by the definition of a Circle AC CB are equal to the Line AB having been divided in the middle in the Point C. And CG being common the Angles GCB GCA would then be equal by the 8th of the 1st and CG would be then a Perpendicular and not CD which would be contrary to the Hypothesis Therefore the Center cannot be out of the Line CD I further add that it must be in the Point F which divideth the same into two equal Parts otherwise the Lines drawn from the Center to the Circumference would not be equal Corollary The Center of a Circle is in a Line which divideth another Line in the middle and that perpendicularly USE THis first Proposition is necessary to demonstrate those which follow PROPOSITION II. THEOREM A Streight Line drawn from one point of the Circumference of a Circle to another shall fall within the same Let there be drawn from the Point B in the Circumference a Line to the Point C. I say that it shall fall wholly within the Circle To prove that it cannot fall without the Circle as BVC having found the Center thereof which is A draw the Lines AB AC AV. Demonstration The Sides AB AC of the Triangle ABC are equal whence by the 5th of the 1st the Angles ABC ACB are equal And seeing the Angle AVC is exteriour in respect of the Triangle AVB it is greater than ABC by the 16th of the 1st it shall be also greater than the Angle ACB Thence by the 19th of the 1st in the Triangle ACV the side AC opposite to the greatest Angle AVC is greater than AV and by consequence AV cannot reach the circumference of the Circle seeing it is shorter than AC which doth but reach the same wherefore the Point V is within the Circle the same may be proved of any Point in the Line AB and therefore the whole Line AB falls within the Circle USE IT is on this Proposition that are grounded those which demonstrate that a Circle toucheth a streight Line but only in one Point for if the Line should touch two Points of its Circumference it would be then drawn from one
inscribed by the little Rectangles through which the Circumference of the Circle passeth and all those Rectangles taken together are equal to the Rectangle AL. Imagine that the Semi-Circle is made to roul about the Diameter EB the Semi-Circle shall describe a hemi-sphere and the inscribed Rectangles will describe inscribed Cylinders in the semi-sphere and the Circumscribed will describe other Cylinders Demonstration The Circumscribed Cylinders surpass more the inscribed than doth the hemi-sphere surpass the same inscribed Cylinders seeing that they are comprehended within the Circumscribed Cylinders Now the Circumscribed surpass the inscribed by so much as is the Cylinder AL therefore the hemi-sphere shall surpass by less the inscribed Cylinders than doth the Cylinder made by the Rectangle AL. The Cylinder AL is less than the Cylinder MP for there is the same Ratio of a great Circle of the Sphere which serveth for Base to the Cylinder AL as of MP to R so then by the foregoing a Cylinder which hath for Base a great Circle of the Sphere and the Altitude R would be equal to the Cylinder MP Consequently the hemi-sphere which surpasseth the quantity D by the Cylinder MP and the inscribed Cylinders by a quantity less than AL surpasseth the inscribed Cylinders by less than the quantity D. Therefore the quantity D is less than the inscribed Cylinders What I have said of a hemi sphere may be said of a whole Sphere LEMMA II. LIke Cylinders inscribed in Two Spheres are in Triple Ratio of the Diameters of the Spheres Lemma Fig. II. If the two like Cylinders CD EF be inscribed in the Spheres A B they shall be in Triple Ratio of the Diameters LM NO Draw the Lines GD IF Demonstration The like Right Cylinders CD EF are like so then there is the same Ratio of HD to DR as of QF to FS as also the same Ratio of KD to KG as of PF to PI. And consequently the Triangles GDK IFP are like by the 6th of the 6th so there shall be the same Ratio of KD to PF as of GD to IF or of LM to ON Now the like Cylinders CD EF are in Triple Ratio of KD and PF the Diameters of their Bases by the 12th therefore the like Cylinders CD EF inscribed in the Spheres A and B are in triple Ratio of the Diameters of the Spheres PROPOSITION XVIII THEOREM SPheres are in triple Ratio of their Diameters The Spheres A and B are in Triple Ratio of their Diameters CD EF. For if they be not in Triple Ratio one of the Spheres as A shall be in a greater Ratio than Triple of that of CD to EF therefore a quantity G less than the Sphere A shall be in Triple Ratio of that of CD to EF and so one might according to the first Lemma inscribe in the Sphere A Cylinders of the same height greater than the quantity G. Let there be inscribed in the Sphere B as many like Cylinders as those of the Cylinder A. Demonstration The Cylinders of the Sphere A to those of the Sphere B shall be in Triple Ratio of that of CD to EF by the preceding Now the quantity G in respect of the Sphere B is in Triple Ratio of CD to EF there is then the same Ratio of the Cylinders of the Sphere A to the like Cylinders of the Sphere B. So then the Cylinders of A being greater than the quantity G the Cylinder B that is to say inscribed in the Sphere B would be greater than the Sphere B which is impossible Therefore the Spheres A and B are in Triple Ratio of that of their Diameters Coroll Spheres are in the same Ratio as are the Cubes of their Diameters seeing that the Cubes being like solids are in Triple Ratio of that of their Sides FINIS ERRATA PAge 14. Line 21. read AFD p. 23. l. 17. for DF r. EF. l. 16. r. DFE l. 27. r. FD. p. 24. l. 1 and 9. for FB r. FD. p. 26. l. 8. r. HI p. 52. l. 14. r. ACB p. 55. l. 22. r. ACF p. 71. l. 3. r. ACB p. 82. l. 19. r. ABC p. 117. l. 13. r. GFE p. 125. l. 9. r. AD. p. 178. l. 16. r. CFD p. 194. l. ult r. AF. p. 197. l. 2. r. CDA p. 218. l. 7. r. ⅓ and ⅓ p. 268. l. 24. r. CE. p. 281. l. 3. r. ECD p. 289. l. 4. r. are simular p. 294. l. 13. r. eight ¾ p. 309. l. 15. r. DBE Advertisement of Globes Books Maps c. made and sold by Philip Lea at the Sign of the Atlas and Hercules in the Poultry near Cheapside London 1. A New Size of Globes about 15 Inches Diameter made according to the more accurate observation and discoveries of our Modern Astronomers and Geographers and much different from all that ever were yet extant all the Southern Constellations according to Mr. Hally's observations in the Island of St. Helena and many of the Nothern Price Four Pounds 2. A size of Globes of about Ten Inches Diameter very much Corrected Price 50 s. 3. Concave Hemispheares Three Inches Diameter which serves as a Case for a Terrestial Globe and may be carried in the Pocket or fitted up in Frames Price 15 or 20 s. 4. Another Globe of about Four Inches Diameter fitted to move in Circular Lines of Brass for Demonstrating the Reason of Dyalling or being erected upon a small Pedestal and fet North and South will shew the hour of the day by its own shadow or by the help of a moving Meridian will shew the hour of the Day in all parts of the World c. 14. There is in the Press a Particular description of the general use of Quadrants for the easie resolving Astronomical Geometrical and Gnomonical Problems and finding the hour and Azimuth universally c. whereunto is added the use of the Nocturnal and equinoctial Dial. 15. A new Map of England Scotland and Ireland with the Roads and a delineation of the Genealogy of the Kings thereof from William the Conqueror to this present time with an Alphabetical Table for the ready finding of the places Price 18 d. 16. The Elements of Euclide explained and demonstrated after a new and most easie method with the uses of each Proposition in all the parts of the Mathematicks by Claude Francois Millet Dechales a Jesuite 17. New Maps of the World four quarters and of all the Countries and of all sizes made according to the latest discoveries extant may be had pasted upon Cloath and Coloured also Sea Plats Mathematical Projection Books and Instruments whatsoever are made and sold by Philip Lea. FINIS EUCLIDS ELEMENTS with the uses EUCLID London Printed for Phillip Lea Globe Maker at the Atlas and Hercules in Cheapside Near Friday Street There all Sorts of Globes Spheres Maps Sea-plats Mathematical Books and Instruments are Made and Sold Plate 2. Propositions and Uses of the first Book See Plate 3. Plate 1. Definitions of the first Book Propositions and Uses of the first Book See Plate 2. Plate 3. Propositions Uses of the first Book Definitions Propositions Uses of the Second Book Plate 4 Definitions of the third Book Propositions Propositions and Uses Plate 5. Definitions Propsitions of the fouth Book Plate 6. Definitions of the Sixth Book Propositions Uses Plate 7. Definitions of the Eleventh Book Propositions Plate 8. Proposi of the Twelfth Book
under AB and AC shall be Three times 8 or 24 the square of AC 3 is 9 the Rectangle comprehended under AC 3 and CB 5 is 3 times 5 or 15. It is evident that 15 and 9 are 24. USE A Â 43 C 40. 3 B Â 3 120. Â 9. 129 Â Â THis Proposition serveth likewise to Demonstrate the ordinary practice of Multiplication For Example if one would Multiply the Number 43 by 3 having separated the Number of 43 into two parts in 40 and 3 three times 43 shall be as much as three times 3 which is Nine the Square of Three and Three times Forty which is 120 for 129 is Three Times 43. Those which are young beginners ought not to be discouraged if they do not conceive immediately these Propositions for they are not difficult but because they do imagine they contain some great Mystery PROPOSITION IV. THEOREM IF a Line be Divided into Two Parts the Square of the whole Line shall be equal to the Two Squares made of its parts and to Two Rectangles comprehended under the same parts Let the Line AB be Divided in C and let the Square thereof ABDE be made let the Diagonal EB be drawn and the Perpendicular CF cutting the same and through that Point let there be drawn GL Parallel to AB It is evident that the Square ABDE is equal to the Four Rectangles GF CL CG LF The Two first are the Square of AC and of CB the Two Complements are comprehended under AC CB. Demonstration The Sides AE AB are equal thence the Angles AEB ABE are half Right and because of the Parallels GL AB the Angles of the Triangles of the Square GE by the 29th shall be equal as also the Sides by the 6th of the 1. Thence GF is the Square of AC In like manner the Rectangle CL is the Square of CB the Rectangle GC is comprehended under AC and AG equal to BL or BC the Rectangle LF is comprehended under LD equal to AC and under FD equal to BC. Coroll If a Diagonal be drawn in a Square the Rectangles through which it passeth are Squares USE A 144 B 22 C 12 THis Proposition giveth us the practical way of finding or extracting the Square Root of a Number propounded Let the same be the number A 144 represented by the Square AD and its Root by the Line AB Moreover I know that the Line required AB must have Two Figures I therefore imagine that the Line AB is Divided in C and that AC representeth the first Figure and BC the Second I seek the Root of the First Figure of the Number 144 which is 100 and I find that it is 10 and making its Square 100 represented by the Square GF I Subtract the same from 144 and there remains 44 for the Rectangles GC FL and the Square CL. But because this gnomonicall Figure is not proper I transport the Rectangle FL in KG and so I have the Rectangle KL containing 44. I know also almost all the Length of the Side KB for AC is 10 therefore KC is 20 I must then Divide 44 by 20 that is to say to find the Divisor I double the Root found and I say how many times 20 in 44 I find it 2 times for the Side BL but because 20 was not the whole Side KB but only KC this 2 which cometh in the Quotient is to be added to the Divisor which then will be 22. So I find the same 2 times precisely in 44 the Square Root then shall be 12. You see that the Square of 144 is equal to the Square of 10 to the Square of 2 which is 4 and to twice 20 which are Two Rectangles comprehended under 2 and under 10. PROPOSITION V. THEOREM IF a Right Line be cut into equal parts and into unequal parts the Rectangle comprehended under the unequal parts together with the Square which is of the middle part or difference of the parts is equal to the Square of half the Line If the Line AB is Divided equally in C and unequally in D the Rectangle AH comprehended under the Segments AD DB together with the Square of CD shall be equal to the Square CF that is of half of AB viz. CB. Make an end of the Figure as you see it the Rectangles LG DI shall be Squares by the Coroll of the 4th I prove that the Rectangle AH comprehended under AD and DH equal to DB with the Square LG is equal to the Square CF. Demonstration The Rectangle AL is equal to the Rectangle DF the one and the other being comprehended under half the Line AB and under BD or DH equal thereto Add to both the Rectangle CH the Rectangle AH shall be equal to the Gnomon LBG Again to both add the Square LG the Rectangle AH with the Square LG shall be equal to the Square CF. ARITHMETICALLY LEt AB be 10 AC is 5 as also CB. Let CD be 2 and DB 3 the Rectangle comprehended under AD 7 and DB 3 that is to say 21 with the Square of CD 2 which is 4 shall be equal to the Square of CB 5 which is 25. USE THis Proposition is very useful in the Third Book we make use thereof in Algebra to Demonstrate the way of finding the Root of an affected Square or Equation PROPOSITION VI. THEOREM IF one add a Line to another which is Divided into Two equal parts the Rectangle comprehended under the Line compounded of both and under the Line added together with the Square of half the Divided Line is equal to the Square of a Line compounded of half the Divided Line and the Line added If one add the Line BD to the Line AB which is equally Divided in C the Rectangle AN comprehended under AD and under DN or DB with the Square of CB is equal to the Square of CD Make the Square of CD and having drawn the Diagonal FD draw BG Parallel to FC which cuts FD in the Point H through which passeth HN Parallel to AB KG shall be the Square of BC and BN that of BD. Demonstration The Rectangles AK CH on equal Bases AC BC are equal by the 38th of the 1st The Complements CH HE are equal by the 43d of the 1st Therefore the Rectangles AK HE are equal Add to both the Rectangle CN and the Square KG the Rectangles AK CN that is to say the Rectangle AN with the Square KG shall be equal to the Rectangles CN HE and to the Square KG that is to say to the Square CE. Arithmetically or by Numbers LEt AB be 8 AC 4 CB 4 BD 3 then AD shall be 11. It is evident that the Rectangle AN three times 11 that is to say 33 with the Square of KG 16 which together are 49 is equal to the Square of CD 7 which is 49 for 7 times 7 is 49. USE Fig. 6. MAurolycus measured the whole Earth by one single
Superficies 18. A Cone is a figure made when one Side of a Right Angled Triangle viz. one of those that contain the Right Angle remaining fixed the Triangle is turned round about till it return to the place from whence it first moved And if the fixed Right Line be equal to the other which containeth the Right Angle then the Cone is a Rectangled Cone but if it be less it is an Obtuse Angled Cone if greater an Acute Angled Cone 19. The Axis of a Cone is that fixed Line about which the Triangle is moved 20. A Cylinder is a figure made by the moving round of a Right Angled Parallelogram one of the sides thereof namely which contains the Right Angle abiding fixed till the parallelogram be turned about to the same place whence it began to move 21. Like Cones and Cylinders are those whose Axes and Diameters of their Bases are Proportional Cones are right when the Axis is perpendicular to the Plain of the Base and they are said to be Scalene when the Axis is inclined to the Base and the Diameter of their Bases are in the same Ratio We add that inclined Cones to be like their Axes must have the same inclination to the Planes of their Bases PROPOSITION I. THEOREM Plate VII Prop. I. A Strait Line cannot have one of its parts in a Plane and the other without it If the Line AB be in the Plane AD it being continued shall not go without but all its parts shall be in the same Plane For if it could be that BC were a part of AB continued Draw in the Plane CD the Line BD perpendicular to AB draw also in the same Plane BE perpendicular to BD. Demonstration The Angles ABD BDE are both Right Angles thence by the 14th of the first AB BE do make but one Line and consequently BC is not a part of the Line AB continued otherwise two strait Lines CB EB would have the same part AB that is AB would be part of both which we have rejected as false in the Thirteenth Maxim of the first Book USE WE establish on this Proposition a principle in Gnomonicks to prove that the shadow of the stile falleth not without the Plane of a great Circle in which the Sun is Seeing that the end or top of the stile is taken for the Center of the Heavens and consequently for the Center of all the great Circles the shadow being always in a streight Line with the Ray drawn from the Sun to the Opaque Body this Ray being in any great Circle the shadow must also be therein PROPOSITION II. THEOREM LInes which cut one another are in the same Plane as well as all the parts of a Triangle If the Two Lines BE CD cut one another in the Point A and if there be made a Triangle by drawing the Base BC I say that all the parts of the Triangle ABC are in the same plane and that the Lines BE CD are likewise therein Demonstration It cannot be said that any one part of the Triangle ABC is in a Plane and that the other part is without without saying that one part of a Line is in one Plane and that the other part of the same Line is not therein which is contrary to the first Proposition and seeing that the sides of the Triangle are in th same Plane wherein the Triangle is the Lines BE CD shall be in the same Plane USE THis Proposition doth sufficiently determine a Plane by two streight Lines mutually intersecting each other or by a Triangle I have made use thereof in Opticks to prove that the objective parallel Lines which fall on the Tablet ought to be Represented by Lines which concur in a Point PROPOSITION III. THEOREM THe common section of two Places is a streight Line If Two Planes AB CD cut one another their common section EF shall be a streight Line For if it were not take Two Points common to both Planes which let be E and F and draw a strait Line from the point E to the point F in the Plane AB which let be EHF Draw also in the Plane CD a streight Line from E to F if it be not the same with the former let it be EGF Demonstration Those Lines drawn in the Two Planes are two different Lines and they comprehend a space whch is contrary to the Twelfth Maxim Thence they are but one Line which being in both Planes shall be their common section USE THis Proposition is fundamental We do suppose it in Gnomonicks when we represent in a Dial the Circles of the hours marking only the common section of their Planes and that of the Wall PROPOSITION IV. THEOREM IF a Line be perpendicular to two other Lines which cut one another it shall be also perpendicular to the Plane of those Lines If the Line AB be perpendicular to the Lines CD EF which cut one another in the point B in such manner that the Angles ABC ABD ABE ABF be right which a flat figure cannot represent it shall be perpendicular to the Plane CD EF that is to say that it shall be Perpendicular to all the Lines that are drawn in that Plane through the point B as to the Line GBH Let equal Lines be cut BC BD BE BF and let be drawn the Lines EC DF AC AD AE AF AG and AH Demonstration The four Triangles ABC ABD ABE ABF have their Angles Right in the Point B and the Sides BC BD BE BF equal with the side AB common to them all Therefore their Bases AC AD AE AF are equal by the 4th of the 1st 2. The Triangles EBC DBF shall be equal in every respect having the Sides BC BD BE BF equal and the Angles CBE DBF opposite at the vertex being equal so then the Angles BCE BDF BEC BFD shall be equal by the 4th of the first and their Bases EC DF equal 3. The Triangles GBC DBH having their opposite Angles CBG DBH equal as also the Angles BDH BCG and the sides BC BD they shall then have by the 26th of the 1st their Sides BG BH CG DH equal 4. The Triangles ACE AFD having their sides AC AD AE AF equal and the Bases EC DF equal they shall have by the 8th of the 1st the Angles ADF ACE equal 5. The Triangles ACG ADH have the Sides AC AD CG DH equal with the Angles ADH AGC Thence they shall have their Bases AG AH equal Lastly the Triangles ABH ABG have all their sides equal thence by the 27th of the 1st the Angles ABG ABH shall be equal and the Line AB perpendicular to GH So then the Line AB shall be perpendicular to any Line which may be drawn through the point B in the Plane of the Lines CD EF which I call perpendicular to the Plane USE THis Proposition cometh often in use in the first Book of Theodosius for example to Demonstrate that the Axis of the World is
Square of the Line added are double to the Square of half the Line and to the Square which is Composed of the half Line and the Line added If one supposeth AB to be Divided in the middle at the Point C and if thereto be added the Line BD the Squares of AB and BD shall be double to the Squares of AC and CD added together Draw the Perpendiculars CE DF equal to AC Then draw the Lines AE EF AG EBG Demonstration The Lines AC CE CB beng equal and the Angles at the Point C being Right The Angles AEC CEB CBE DBG DGB shall be half Right and the Lines DB DG and EF FG CD shall be equal The Square of AE is double to the Square of AC the Square of EG is double to the Square of EF or CD by the 47th of the 1st Now the Square of AG is equal to the Squares of AE EG by the 47th of the 1st Therefore the Square of AG is double to the Squares of AC CD The same AG by the 47th of the 1st is equal to the Squares of AD BD or GD Therefore the Squares of AD BD are double the Squares of AC CD ARITHMETICALLY LEt AB be 6 parts AC 3 CB 3 BD 4 the Square of AD 10 is 100. the Square of BD 4 is 16 which are 116. The Square of AC 3 is 9 the Square of CD 7 is 49. Now 49. and 9 is 58 the half of 116. PROPOSITION XI PROBLEM TO Divide a Line so that the Rectangle comprehended under the whole Line and under one of its parts shall be equal to the Square of the other part It is proposed to Divide the Line AB so that the Rectangle comprehended under the whole Line AB and under HB be equal to the Square of AH Make a Square of AB by the 46th of the 1st Divide AD in the middle in E then Draw EB and make EF equal to EB Make the Square AF that is to say that AF AH be equal I say that the Square of AH shall be equal to the Rectangle HC comprehended under HB and the Line BC equal to AB Demonstration The Line AD is equally divided in E and there is added thereto the Line FA thence by the 6th the Rectangle DG comprehended under DF and FG equal to AF with the Square of AE is equal to the Square of EF equal to EB Now the Square of EB is equal to the Squares of AB AE by the 47th of the 1st therefore the Squares of AB AE are equal to the Rectangle DG and to the Square of AE and taking away from both the Square of AE the Square of AB which is AC shall be equal to the Rectangle DG taking also away the Rectangle DH which is in both the Rectangle HC shall be equal to the Square of AG. USE THis Proposition serveth to cut a Line in extream and mean Proportion as shall be shewn in the Sixth Book It is used often in the 14th of Euclid's Elements to find the Sides of Regular Bodies It serveth for the 10th of the Fourth Book to inscribe a Pentagone in a Circle as also a Pentadecagone You shall see other uses of a Line thus divided in the 30th of the Sixth Book PROPOSITION XII THEOREM IN an obtuse angled Triangle the Square of the side opposite to the obtuse Angle is equal to the Squares of the other two sides and to two Rectangles comprehended under the side on which one draweth a Perpendicular and under the Line which is between the Triangle and that perpendicular Let the Angle ACB of the Triangle ABC be obtuse and let AD be drawn perpendicular to BC. The Square of the side AB is equal to the Squares of the sides AC CB and to two Rectangles comprehended under the side BC and under DC Demonstration The Square of AB is equal to the Squares of AD DB. by the 47th of the 1st the Square of DB is equal to the Squares of DC and CB and to two Rectangles comprehended under DC CB by the 4th therefore the Square of AB is equal to the Squares of AD DC CB and to two Rectangles comprehended under DC CB in the place of the two last Squares AD DC Put the Square of AC which is equal to them by the 47th of the 1st the Square of AB shall be equal to the Square of AC and CB. and to two Rectangles comprehended under DC CB. USE THis Proposition is useful to measure the Area of a Triangle it s three sides being known for Example If the side AB was twenty Foot AC 13 BC 11 the Square of AB would be four hundred the Square of AC one hundred sixty nine and the Square of BC one hundred twenty one the Sum of the two last is Two hundred and ninety which being subtracted from four hundred leaves one hundred and ten for the two Rectangles under BC CD the one half-fifty five shall be one of those Rectangles which divided by BC 11 we shall have five for the Line CD whose Square is twenty five which being subtracted from the Square of AC one hundred sixty nine there remains the Square of AD one hundred forty four and its Root shall be the side AD which being multiplied by 5½ the half of BC you have the Area of the Triangle ABC containing 66 square Feet PROPOSITION XIII THEOREM IN any Triangle whatever the Square of the side opposite to an acute Angle together with two Rectangles comprehended under the side on which the Perpendicular falleth and under the Line which is betwixt the Perpendicular and that Angle is equal to the Square of the other sides Let the proposed Triangle be ABC which hath the Angle C acute and if one draw AD perpendicular to BC the Square of the side AB which is opposite to the acute Angle C together with two Rectangles comprehended under BC DC shall be equal to the Squares of AC BC. Demonstration The Line BC is divided in D whence by the 7th the Square of BC DC are equal to two Rectangles under BC DC and to the Square of BD add to both the Square of AD the Square of BD DC AD shall be equal to two Rectangles under BC DC and to the Squares of BD AD in the place of the Squares of CD AD put the Square of AC which is equal to them by the 47th of the 1st and instead of the Squares of BD AB substitute the Square of AB which is equal to them the Squares of BC AC shall be equal to the Square of AB and to two Rectangles comprehended under BC DC USE THese Propositions are very necessary in Trigonometry I make use thereof in the eighth Proposition of the third Book to prove That in a Triangle there is the same Reason between the whole Sine and the Sine of an Angle as are between the Rectangle of the sides comprehending that Angle and
of 4 Foot and thrice in the Line of 6 the first to the second shall have the same Reason as 2 to 3. Irrational Reason is between Two Magnitudes of the same Species which are incommensurable that is to say that have not a common measure As the Reason of the Side of a Square to its Diagonal For there cannot be found any measure although never so little which will measure both precisely Four Magnitudes shall be in the same Reason or shall be Proportionals when the Reason of the first to the second shall be the same or like to that of the third to the fourth wherefore to speak properly Proportion is a similitude of Reason But one findeth it difficult to understand in what consisteth this similitude of Reason It is only to say that two habitudes or Relations be alike For Euclid hath not given a just Definition and which might have explained its Nature having contented himself to give us a mark by which we may know if Magnitude have the same Reason And the obscurity of this Definition hath made this Book difficult I will endeavour to supply this default 5. Euclide saith that Four Magnitudes have the same Reason when having taken the Equi-multiplices of the first and of the third and other Equimultiplices of the second and of the fourth whatever combination is made when the Multiplex of the first is greater than the Multiplex of the second the Multiplex of the third shall be also greater than the Multiplex of the fourth And when the Multiplex of the first is equal or less than the Multiplex of the second the Multiplex of the third is equal or less than the Multiplex of the fourth That then there is the same Reason between the first and second as there is between the third and fourth A B C D 2. 4. 3. 6. E F G H 10 8. 15 12. K L M N 8. 8. 12. 12. O P Q R 6. 16 9. 24 As if there were proposed four Magnitudes A B C D. Having taken the Equi-multiplexes of A and C which let be E and G quintuplex F and H double to B and D. In like manner taking K and M quadruple to A and C L and N double to B and D. Taking again O and Q triple to A and C P and R quadruple to B and D. Now because E being greater than F G is greater than H and K beng equal to L M is equal to N In fine O being lesser than P Q is lesser than R. Then A shall have the same Reason to B as C to D. I believe that Euclid ought to have Demonstrated this Proposition seeing it is so intangled that it cannot pass for a Maxim To explain well what Proportion is it is to say that four Magnitudes have the same Ratio although one may say in general that to that end the first must be alike part or a like whole in respect of the second as is the third compared to the fourth notwithstanding because this Definition doth not convene with the Reason of equality there must be given a more general and to make it intelligible it must be explained what is meant by a like Aliquot part Like Aliquot parts are those which are as many times in their whole as three in respect of nine two in respect of six are alike Aliquot parts because each are found three times in their whole The first quantity will have the same Reason to the second as the third hath to the fourth if the first contains as many times any Aliquot part of the second whatever as the 3d. contains alike Aliquot part of the 4th A B C D. as if A contains as many times a Hundreth a Thousandth a Millionth part of B as C contains a Hundreth a Thousandth a Millionth part of D and so of any other Aliquot parts imaginable there will be the same Reason of A to B as of C to D. To make this Definition yet clearer I will in the first place prove that if there be the same reason of A to B as there is of C to D A will contain as many times the Aliquot parts of B as C doth of D. And I will afterwards prove that if A contains as many times the Aliquot parts of B as C doth of D there will be the same Reason of A to B as of C to D. The first Point seemeth evident enough provided one doth conceive the terms for if A contains one Hundred and one times the tenth part of B and C only One Hundred times the tenth part of D the Magnitude A compared with B would be a greater whole than C compared with D so that it could not be compared after the same manner that is to say the habitude or Relation would not be the same The second point seemeth more difficult to wit whether if this propriety be so found the Reason shall be the same that is to say if AB contains as many times any Aliquot parts whatever of CD as E contains like Aliquot parts of F there shall be the same Reason of AB to CD as of E to F. For I will prove that if there were not the same Reason A would contain more times any Aliquot part of B than C containeth alike Aliquot parts of D which would be contrary to what we had supposed Demonstration Seeing there is the same Reason of AG to CD as of E to F AG will contain as many times KD an Aliquot part of CD as E would contain a like Aliquot part of F. Now AB contains KD once more than AG thence AB will contain once more KD an Aliquot part of CD than E doth contain a like Aliquot part of F which would be contrary to the supposition 6. There will be a greater Reason of the first quantity to the second than of the third to the fourth if the first contains more times any Aliquot part of the second than the third doth contain a like Aliquot part of the fourth As 101 hath a greater Reason to 10 than 200 to 20 because that 101 contains One Hundred and one times the Tenth part of 10 and 200 contains only One Hundred times the Tenth part of 20 which is 2. 7. The Magnitudes or quantities which are in the same Reason are called Proportionals 8. A Proportion or Analogie is a Similitude of Reason or habitude 9. A Proportion ought to have at least three terms For to the end there be similitude of Reason there must be two Reasons Now each Reason having two terms the antecedent and the consequent it seemeth there ought to be four as when we say that there is the same Reason of A to B as of C to D but because the consequent of the first Reason may be taken for antecedent in the second three terms may suffice as when I say that there is the same Reason of A to B as of B to C. 10. Magnitudes are in continued Proportion
when the Terms between them are taken twice that is to say as antecedent and as consequent As if there be the same Reason of A to B as of B to C and of C to D. 11. Then A to C shall be in duplicate Ratio of A to B and the Ratio of A to D shall be in triplicate Ratio to that of A to B. It is to be taken notice that there is a great deal of difference between double Ratio and duplicate Ratio We say that the Ratio of four to two is double that is to say four is the double of two whence it followeth that the number two is that which giveth the Name to this Ratio or rather to the Antecedent of this Ratio So we we say double triple quadruple quintuple which are Denominations taken from those numbers duo tres quatuor quinque compared with unity for we better conceive a Reason when its terms are small But as I have already taken notice those Denominations fall rather on the Antecedent than on the Reason it self we call that double triple Reason or Ratio when the Antecedent is double or triple to the consequent but when we say the Reason is duplicate we mean a Reason compounded of two like Reasons as if there be the same Reason of two to four as of four to eight the Reason of two and eight being compounded of the Reason of two and four and of that of four and eight which are alike and as equal the Reason or Ratio of two to eight shall be duplicated by each Three to twenty seven is a duplicated Reason of that of three to nine The Reason of two to four is called subduple that is to say two is the half of four but the reason of two to eight is duple of the sub-duple that is to say that two is the half of the half of eight as three is the third of the third of twenty seven where you see there is taken twice the Denominator ½ and ⅓ In like manner eight to two is a duplicate reason of eight to four because eight is double to four but eight is the double of the double of two If there be four terms in the same continued Reason that of the first and last is triple to that of the first and second as if one put these four Numbers two four eight sixteen the reason of two to sixteen is triple of two to four for two is the half of the half of the half of sixteen As the reason of sixteen to two is triple of sixteen to eight for sixteen is the double of eight and it is the double of the double of the double of two 12. Magnitudes are homologous the Antecedents to the Antecedents and the Consequents to the Consequents As if there be the same Reason of A to B as of C to D A and C are homologous or Magnitudes of a like Ratio The following Definitions are ways of arguing by Proportion and it is principally to demonstrate the same that this Book is composed 13. Alternate Reason or by Permutation or Exchange is when we compare the Antecedents one with the other as also the consequents For example if because there is the same reason of A to B as of C to D I conclude there is the same reason of A to C as of B to D this way of reasoning cannot take place but when the four terms are of the same Specie that is to say either all four Lines or Superficies or Solids Proposition 16. 14. Converse or Inverse Reason is a comparison of the Consequents to the Antecedents As if because there is the same reason of A to B as of C to D I conclude there is the same reason of B to A as there is of D to C. Proposition 16. 15. Composition of Reason is a comparison of the Antecedent and Consequent taken together to the Consequent alone As if there be the same Reason of A to B as of C to D I conclude also that there is the same reason of AB to B as of CD to D. Prop. 18. 16. Division of Reason is a comparison of the excess of the Antecedent above the Consequent to the same Consequent As if there be the same reason of AB to B as of CD to D I conclude that there is the same reason of A to B as of CD Prop. 17. 17. Conversion of Reason is the comparison of the Antecedent to the difference of the Terms As if there be the same reason of AB to B as of CD to D I conclude that there is the same reason of AB to A as of CD to C. Proposition 18. 18. Proportion of Equality is a comparison of the extream Quantities in leaving out those in the middle A B C D E F G H. As if there were the same reason of A to B as of E to F and of B to C as of F to G and of C to D as of G to H. I draw this Consequence that there is therefore the same reason of A to D as of E to H. 19. Proportion of Equality well ranked is that in which one compareth the Terms in the same manner of Order as in the preceding Example Prop. 22. 20. Proportion of Equality ill ranked is that in which one compareth the Terms with a different Order As if there were the same reason of A to B as of G to H and of B to C as of F to G and of C to D as of E to F. I draw this Conclusion that there is the same reason of A to D as of E to H. Prop. 28. Here is all the ways of arguing by Proportion There is the same reason of A to B as of C to D therefore by alternate reason there is the same reason of A to C as of B to D and by inversed reason there is the same Reason of B to A as of D to C and by composition there is the same reason of AB to B as of CD to D. By Division of Reason if there be the same reason of AB to B as of CD to D there is the same reason of A to B as of C to D and by Conversion there is the same reason of AB to A as of CD to C. By reason of Equality well ranked if there be the same reason of A to B as of C to D and also the same reason of B to E as of D to F there will be the same reason of A to E as of C to F. By reason of Equality ill ranked if there be the same reason of A to B as of D to F and also the same reason of B to E as of C to D there will be the same reason of A to E as of C to F. This Book contains twenty five Propositions of Euclid to which there has been added ten which are received The first six of this Book are useful only to prove the following Propositions by the method