point C. to the point F. Draw the lines F C F D. C D F shall be the Triangle required Of the Exagone Bring six times the half Diameter A B. within the circumference given Of the Dodecagone Cut the Arch of the Exagone A C. into two equally in O. A O shall be the side of the Dodecagone Here ended the Ten of Hearts See the King of Hearts PROPOSITION II. Within a Circle given to inscribe a Square and an Octogone Let A B C D be the Circle within the which one would inscribe a Square and an Octogone The Practice Of the Square Draw the two Diameters A B C D. dividing each other at Right Angles that is to say draw the right line C D. by the center of the Circle O. from the Points or Ends C and D. Make the Sections I and L. draw the right line I L. passing also by the Center O. the Lines or Diameters A B C D. shall divide themselves at right Angles being the lines A C A D B C B D. And A C B D shall be the Square required Of the Octogone Subdivide every fourth of the Circle you shall make the Octogone PROPOSITION III. Within a Circle given to inscribe a Pentagone and a Decagone Let A B C D be the Circle propounded The Practice Of the Pentagone Draw the two Diameters A B C D. dividing themselves at right Angles in E. Divide the half Diameters C E. into two equally in F. from the point F. and from the Interval F A. Describe the Arch A G. from the point A. and from the Interval A G. describe the Arch G H. The righ line A H. shall divide the circle into five equal parts Of the Decagone Subdivide every part of the circle into two equally Here endeth the King of Hearts See the Queen of Hearts PROPOSITION IV. Within a Circle given to inscribe an Eptagone Let A B C be the circle propounded within the which we must make an Eptagone The Practice Draw the half Diameter I A. from the end A. and from the Interval A I. describe the arch C I C. draw the right line C C. bear the half C O. seven times within the circumference of the circle you shall have the Eptagone required PROPOSITION V. Within a Circle given to describe an Enneagone Let B C D be a circle propounded within which one would inscribe an Enneagone The Practice Draw the half Diameter A B. from the end B. and from the Interval B A. Describe the Arch C A D. draw the right line C D. enlarged towards F. make the line E F. equal to the line A B. from the point E. Describe the Arch F G. from the point F. Describe the Arch E G. draw the right line A G. D H shall be the Ninth part of the circumference Here endeth the Queen of Hearts See the Knave of Hearts PROPOSITION VI. Within a Circle given to inscribe an Endecagone Let A E F be the Circle given within which we must inscribe an Endecagone The Practice Draw the half Diameter A B. divide the half Diameter A B. into two equally in C. from the points A and C. and from the Interval A C. Describe the Arches C D I A D. from the point I. and from the Interval I D. Describe the Arch D O. the Interval C O. shall be the side of the Endecagone required very punctually PROPOSITION VII Within the Circle given to inscribe such a Poligone as one would The Practice Draw the Diameter A B. describe the circle A B F. capable to contain seven times A B. as if you would frame upon A B. a Poligone like to that which you should inscribe within the circle given A B C. Draw the Diameter D E. parallel to the Diameter A B. draw the right lines D A G F B H. by the ends D A E B. G H shall divide the circle given A B C. into seven equal parts And so of all other Poligones Here endeth the Knave of Hearts See the Ace of Spades PROPOSITION VIII From a Circle given to take a Portion Capable of an Angle equal to an Angle Rectilinear propounded Let A C E be the Circle given from which we must take a Portion capable to contain an Angle equal to the Angle D. The Practice Draw the half Diameter A B. bring the line touching A F. make the Angle F A C. equal to the Angle given D. All the Angles which shall be framed upon the line A C. and within the portion A E C. shall be all equal to the Angle given D. so the portion A E C. is the required PROPOSITION IX Within a Circle to inscribe a Triangle of equal Angles to a Triangle given Let A B C be the Circle within the which we must inscribe a Triangle like to the Triangle D E F. The Practice Bring the line touching G H. from the point of the touching A. make the Angle H A C. equal to the Angle E. make also the Angle G. A B. equal to the Angle D. Draw the line B C. A B C is the Triangle required like to the Triangle given D E F. PROPOSITION X. To inscribe a Circle within a Triangle given Let A B C be the Triangle within the which we must inscribe a circle The Practice Divide the two Angles B and C. each into two equally by the right lines B D C D. From the Section D. bring down the perpendicular D F. From the Section or Center D. and from the Interval D F. describe the circle demanded E F G. Here endeth the Ace of Spades See the Deux of Spades PROPOSITION XI To inscribe a Square within a Triangle given Let A B C be the Triangle within the which we must inscribe a square The Practice Elevate the perpendicular A D. at the end of the Basis A B. make this perpendicular A D. equal to the basis A B. From the Angle C. draw the line C E. parallel to the line A D. Bring the oblique line D E. from the section F. draw the line F G. parallel to the basis A B. draw the lines F H G I. parallel to the line C E F G H I shall be the Square required PROPOSITION XII To inscribe a Pentagone Regular within a Triangle Equilateral Let A B C be the Triangle within the which one would inscribe a Pentagone The Practice Bring down the Perpendicular A I. from the center A. describe the Arch B I M. divide into five equal parts the Arch B I. bring the sixth I M. draw the line A M. divide A M. into two equally in L. from the point A. describe the Arch L D. draw the right line L D unto H. Make the part A G. equal to the part B H. draw the right line D G M C. from the center D. and from the Interval of the section N. describe the Arch N O. from the points N and O. describe the Arches D Q D P. draw the lines O P
P Q N Q. D O P Q N shall be the Pentagone required PROPOSITION XIII To inscribe a Triangle Equilateral within a square Let A B C D be the square within the which we must make a Triangle Equilateral The Practice Draw the Diagonals A C B D. from the Center E. and from the Interval E A. describe the circle A B C D. from the point C. and the Interval C E. Describe the Arch G E F. draw the right lines A F A G. bring the right line H I. A H L shall be the Triangle Equilateral required Here endeth the Deux of Spades See the Trois of Spades PROPOSITION XIV To inscribe a Triangle Equilateral within a Pentagone Let A B C D E be the Pentagone within the which we must inscribe a Triangle Equilateral The Practice Circumscribe the circle A B C D E. from the point A. and from the Interval of the half Diameter A F. describe the Arch F L. divide this Arch F L. into two equally in N. draw the line A N I. from the point A. and from the Interval A I. describe the Arch I O H. draw the lines A H H I. A H I shall be the Triangle demanded PROPOSITION XV. To inscribe a Square within a Pentagone Let A B C D E be the Pentagone within the which we must inscribe a square The Practice Draw the right line B E. let down the Perpendicular E T. at the end of B E. make this Perpendicular E T. equal to the line B E. draw the line E T. from the section O. bring the line O P. parallel to the side C D. at the end O P. Elevate the perpendiculars O M P I. draw the line N M. N M O P shall be the square required Here endeth the Trois of Spades See the Four of Spades PROPOSITION I. About a Triangle given to Circumscribe a Circle LEt A B C be the Triangle about the which one would circumscribe a Circle The Practice Describe the circumference A B C. by the three Points A B C. and you shall have the demanded PROPOSITION II. About a Square to Circumscribe a Circle Let A B C D be the Square about which we must circumscribe a Circle The Practice Draw the two Diagonals A B C D. from the section or Center G. and from the Interval G A. describe the circle demanded A B C D. PROPOSITION III. About a Circle to circumscribe a Triangle of equal Angles to a Triangle given Let D E V be the Circle about the which we must make a Triangle which may be like to the Triangle F G H. The Practice Draw the Diameter A B. by the center C. make the Angle A C E. equal to the Angle H. make the Angle B C D. equal to the Angle G. prolong the lines E C D C. towards R and S. draw the line Tangent N O. parallel to the line D R. draw the line Tangent O I. parallel to the line E S. draw also the line Touchant N I. parallel to the Diameter A B. I N O shall be the Triangle demanded like to the Triangle F G H circumscribed about the circle D E V. Here endeth the Four of Spades See the Five of Spades PROPOSITION IV. About a Circle to circumscribe a Square Let A B C D be the circle about the which we must describe a Square The Practice Draw the Diameters A B C D. dividing themselves at right Angles in O. From the points A C B D and from the Interval A O. describe the Demicircles H O G H O E E O F F O G draw the right lines E F F G G H H E by the Section E F G H. E F G H shall be the square demanded PROPOSITION V. About a Circle given to circumscribe a Pentagone The Practice Let A B C D E be the Circle given about the which one would describe a Pentagone Inscribe the Pentagone A B C D E. from the center F. and by the midst of each of the sides draw the lines F O F P F Q F R F S. bring the line F A. draw the line Tangent P Q. by the point A. from the center F. and from the Interval F. P. describe the circle O P Q R S. draw the sides of the Pentagone demanded by the Section O P Q R S. PROPOSITION VI. About a Poligone Regular to circumscribe the same Poligone Let B C D E F G be the Poligone given about the which we must circumscribe another Poligone like The Practice Prolong two sides as B G E F. unto the point of the meeting H. draw the line A H. draw the line F I. dividing the Angle G F H. into two equally from the center A. and from the Interval A I. describe the circle I M O. draw the Rays A L A M A N A O by the midst of each sides draw the sides of the outward Poligone demanded by the Sections I L M N O P. PROPOSITION VII About a Triangle Equilateral to Circumframe a Square Let A B C be a Triangle Equilateral about the which we must circumscribe a Square The Practice Divide the Basis B C. into two equally in E. prolong this Basis B C. the one part and the other towards D and D. Make the Lines E D E D. equal to the line E A. From the point E. and from the Interval E C. describe the Demy-circle B F C. draw the line A E F. from the point F. draw the lines F C G F B G. A G F G shall be the Square demanded Here endeth the Five of Spades See the Sixth of Spades PROPOSITION VIII About a Triangle given Equilateral to circumscribe a Pentagone Let A B C be the Triangle given about the which we must describe a pentagone The Practice From the Point or Angle A B C. and with the same opening of the Compasses describe at discretion the Arch D E L P. divide the Arch D O. into five equal parts 1 2 3 4 5. From the center or section O. and from the Interval of 4 parts O N. describe the Arch N M E. draw the right line A E F. divide the Arch M P. equal to the Arch E N. draw the right line F P C G. equal to the line F A. make the arch D H. equal to the Arch D E. draw the sides A I I R. equal to the sides A F F G. the side G R shall finish the Pentagone demanded PROPOSITION IX About a Square to circumscribe a Triangle of equal Angles to a Triangle given Let D E F G be the Square about the which we must circumscribe a Triangle like to the Triangle A B C. The Practice Make the Angle E F M. equal to the Angle A. make the Angle M E F. equal to the Angle B. prolong the lines M E M F D G towards I and H. M I H shall be the Triangle required like to the Triangle A B C. and circumscribe about the square given D
hollow part it is called a Concave if it be plane and united it is called a Plane B. A Convex Superficies C. A Concave Superficies A. A Plane Superficies The first Part teacheth only the Construction a framing of plain Superficies A Term or bound is the extremity of any thing The Point is the Term or bound of the Line The Line is the Term of the Superficies And the Superficies is the Term or bound of a Body Here endeth the Four of Diamonds See the Five of Diamouds Of Superficies or Figures Rectilinear THe Superficies do take particular Names according to the Number of their Sides as A. A Trigone or Triangle a Figure of 3 sides B. A Tetragone or Square a Figure of 4 sides C. A Pentagone or Figure of 5 sides D. An Exagone or Figure of 6 sides E. Eptagone a Figure of 7 sides F. Octogone a Figure of 8 sides G. Enneagone a Figure of 9 sides H. Decagone a Figure of 10 sides I. Endecagone a Figure of 11 sides L. Dodecagone a figure of 12 sides All these Figures are called likewise by one general Name Poligones Of TRIANGLES The Triangles are also distinguished by the quality of their Angles and by the disposition of their sides as M. A Triangle Rectangle which hath a Right Angle N. A Triangle Ambligone which hath an Obtuse Angle O. A Triangle Oxigone which hath Three Angles sharp P. A Triangle Equilateral which hath its Three sides equal Q. A Triangle Iso fele which hath two sides equal only R. A Triangle Scalene which hath his three sides unequal Here endeth the Five of Diamonds See the Six of Diamonds Of Figures of Four Sides A. THe Square is a Figure composed of four equal Sides and four right Angles B. A Long Square is a Superficies Rectangle that is to say which hath its Angles right but hath not its Sides equal A B C. A Parallelo-gramme is a Square-side figure whereof the opposite sides are Parallels D. A Rhombus or Lozange is a square side figure which hath the four sides equal but not the four Angles E. A Rhomboid which hath the Angles and the side opposite equal without being equal-angled or equal-sided F. A Trapeze which hath only two sides opposite Parallel and the other two equal G. A Trapezoide or Tablett which hath its Sides and its Angles unequal H. A Gnomon is the excess of a Parallelo-gram upon another Parallo-gram framed upon the same Diagonal All other Figures of more than four Sides are called by one general Name Multi-lateres or Many-Sizes Here endeth the Six of Diamonds See the Seven of Diamonds Of Figures Crooked or Curvi-linear A. A Circle is a Superficies or Figure perfectly round described or drawn from a Center from which the whole Circumference is of equal distance A B C D. The Circumference is the Extremity or outmost part of the Circle otherwise it is the Circular-line that encloseth it B. An Oval is a crooked Figure drawn from many Centers and which all the Diameters divide into two equally C. An Eclipse is also a crooked Figure drawn from many Centers but in shape of an Egg within the which there is but one only Diameter which divideth it into two equally D. A Volute is a Figure or Superficies encompassed by a Line Spiral Of FIGURES Composite A. A Demi-Circle is a Figure contained in the Diameter with the half of the Circumference B. A part of a Circle is a Figure contained within a right Line and a part of a Circle F. The great Portion of the Circle is that which containeth more than the half of the Circle G. The small Portion of a Circle is that which containeth less than the half of the Circle C. A Sector is a Figure contained within two Semi-Diameters with more or less than the half of the Circle There is likewise the great and the small Sector D. Figures Concentrical are those which have one and the same Center E. Figures Excentrical are those which are contained in others of divers Centers Here endeth the Seven of Diamonds See the Eight of Diamonds Of Figures Regular and Irregular A. A Figure Regular it that which hath its opposite parts like an Equal B. An Irregular Figure is that which is composed of Angles and Sides unlike E E. Figures a-like are those whereof all the Parts of one are proportionable to all the Parts of the other although the one be greater or equal or lesser than the other F F. Figures equal are those which contain equally which may be like and unlike C. The Figure Equi-Angle which hath all its Angles equal E E. One Figure is Equi-Angle to another when as all the Angles of the one are equal to all the Angles of the other C D. A Figure Equilateral which hath all its Sides equal Here endeth the Eight of Diamonds See the Nine of Diamonds The AXIOMES I. THings equal to the one and the same are equal amongst themselves The Lines A C A C. which are equal to A B. By the Definition of a Circle are also equal between themselves II. If to equal Things one shall add Things equal all will become equal The Lines A C A C are equal The added C D C D are equal All of them A D A D. are also equal III. If from equal Things one take equal Things the Remainder shall be equal If from equal Lines A D A D. One take equal Parts A C A C The Parts remaining C D C D. Shall be also equal IV. If to Things unequal one add Things equal the whole will be unequal If two Lines unequal D E D E. One does add the equal A D A D. The whole A E A E. Shall be unequal V. If from Things unequal one take Things equal ehe Remainder shall be unequal If from the Lines unequal A E A E. One take the equal A D A D. The remainder D E D E. Shall be unequal VI. The Things that are double to one another are equal between themselves The right Lines D D D D. Which are double to the line A D. Are equal between themselves VII The Things which are the half of the one and the same or of Things equal are equal amongst themselves The Lines A D A D which are half of the Lines D D D D are equal between themselves That which is said of Lines may be also said of Numbers Superficies and of Bodies Here endeth the Nine of Diamonds See the Ten of Diamonds The Petitions or Demands serving for the Ordering of the Practice Demand I. DRaw a straight Line from the Point A. to the Point B. The Practice Apply the Rule to the Points A and B. Draw the Line demanded A B. by letting the Pencil or Draught run close to the rule from the point A. unto the point B. Demand II. Enlarge infinitely the Line C D. as the side of the end D. The Practice Join the Rule to the Line C D. Continue infinitely the said Line C D.
on the side of the end D. letting the Pen run by the Ruler towards E. Demand III. Draw a Circle from the Point A. and the Interval A B. The Practice Set one of the Points of the Compass at the Point given A. Open the other unto the point B. Turn the Compass upon the Point A. And drawing it from the Point B. Describe the Circle demanded B C D. Demand IV. From the Points given E and F. make a Section The Practice Open the Compass at pleasure yet in such manner that the opening of two Points may be greater than the half of the distance which is between the two Points propounded E and F. By this opening of the Compass from the Point E draw the Arch L M. From the point F. draw the Arch H I. The Section G. shall be the demanded Here endeth the Ten of Diamonds See the King of Diamonds PROPOSITION I. TO elevate a Perpendicular from a Point propounded within the midst of a straight Line The Position Let C be the Point propounded within the midst of the Line A B from which a Perpendicular must be elevated The Practice From the Point given C. draw at pleasure the half-circle D E from the Points D and E. Make the Section I. from the Point C. Draw the right Line demanded C O. by Section I. This Line C O. shall be Perpendicular to the Line given A B. and elevated from the Point propounded C. PROPOSITION II. To elevate a Perpendicular at the end of a right Line propounded Let A be the end propounded of the Line A B. upon the which a Perpendicular must be elevated The Practice Set at pleasure the Point C. above the Line A B. From this Point C. and the Interval C A. draw the portion of the Circle E A D. Bring the right Line D C E. by the Points D and C. Draw the Line demanded A E. it shall be Perpendicular to A B. and to the End propounded A Otherwise From the Point A draw the Arch G H M. From the Point G draw the Arch A H. From the Point H draw the Arch A M N. From the Point M draw the Arch H N. Draw the Line demanded A N. Here endeth the King of Diamonds See the Queen of Diamonds PROPOSITION III. Upon an Angle given to elevate a right Line which inclineth neither to the right hand nor to the left The Position Let B A C be the Angle upon the which a right Line must be elevated which inclineth neither to the right hand nor left The Practice From the Angle given A. draw at pleasure the Arch B C. from the Points or Ends B and C. Make the Section D. from the Point or Angle given A. Draw the right Line required A D. by the Section D. This right Line A D. shall be elevated upon the Angle B A C. without inclining either to the right hand or to the left PROPOSITION IV. To depress or bring down a Perpendicular-Line upon a right Line given and from a Point without the same The Position Let C. be the Point given from which a Perpendicular Line is to be brought down upon the Line A B. The Practice From the Point given C. draw at pleasure the Arch D E. cutting the Line A B. at the Points D and E. From the Points D and E. make the Section F. Draw the Line C F. The Line C O. will be the Line Demanded PROPOSITION V. By a Point given to draw a Line Parallel to a right Line given The Position Let A be the Point by the which we must draw a Line which may be Parallel to the Line B C. The Practice Draw at pleasure the oblique Line A D. from the Point A. Draw the Arch D E. from the Point D. Draw the Arch A F. Make the Arch D G. equal to the Arch A F. Bring the Line demanded M N. by the Points A and G. Otherwise From the Point A. draw the Arch E F G. touching the Line B C. Without changing the opening of the Compasses From the Point H. draw the Arch L R I. The Point H. is placed at pleasure within the line B C. Draw the Line demanded O P. by the Point A. and grating upon the Arch L R I. Here endeth the Queen of Diamonds See the Knave of Diamonds PROPOSITION VI. To cut a right Line given and bounded into two equally The Position Let A B. be the right Line propounded to be cut into two equally The Practice From the point or end A. draw the Arch C D. Without changing the opening of the Compasses From the point or end B. draw the Arch E F. These two Arches must divide each other Draw the right Line G H. by the Sections G and H. A B shall be divided into two equally at the Point O. PROPOSITION VII To cut an Angle Rectilinear given into two equally The Position Let B A C. be the Angle propounded to be cut into two equally The Practice From the Angle A. draw at pleasure the Arch D E. From the Points D and E. make the Section O. Draw the Line A O. This Line A O. shall divide the Angle given B A C. into two equally Here endeth the Knave of Diamonds See the Ace of Harts PROPOSITION VIII At the end of a right Line to make an Angle Rectilinear equal to an Angle Rectilinear proposed The Position Let A. be the end of the Line A B. to which we must make an Angle equal to the Angle Rectilinear given C D G. The Practice From the Angle D. draw at pleasure the Arch C G. Without changing the opening of the Compasses From the Point or End A. draw the Arch H O. Make the Arch H E. equal to the Arch C G. Draw the Line A E. The Angle B A E. shall be equal to the Angle C D G. the which was propounded to be done PROPOSITION IX To divide a straight Line given into as many equal Parts as one will The Position Let A B. be the Line proposed to be divided into Six equal Parts The Practice From the end A. draw at discretion the Line A C. From the end B. draw the Line B D. parallel to the Line A C. from the Points A and B. and upon the Lines A C B D. Bring at discretion six equal Parts viz. E F G H I L. upon the Line A C. R Q P O N M upon the Line B D. Draw the Lines E N. F O. G P. H Q. L R. The Line A B. shall be divided into six equal Parts by the Sections S. T. V. X. Y. PROPOSITION X. From a Point given to draw a straight Line which toucheth a Circle propounded The Position Let A. be the Point from which we must draw a Line which toucheth the Circle D O P. The Practice From the Center of the Circle B. draw the Line Secant B A. Divide this Line B A. into two equally in C. from the point C. and the Interval C A. Draw the
one would frame any Poligone The Practice Divide the Arch A C. into twelve equal parts from the point C. Take as many parts upon C A. as there is need above the Twelve to have as many parts as one requireth of the Sides Example If you make a Figure of fifteen Sides From the point C. and from the Interval of three parts C E. describe the Arch E O. A C of 12 C O of 3 will make together 15. From the point O and the Interval O B. describe the Arch B E. From the point F. and the Interval F A. describe a Circumference it shall contain fifteen times the Line given A B. And so of other Poligones Here endeth the Six of Harts See the Seven of Harts PROPOSITION VIII Upon a right Line given to describe a Portion of a Circle capable of an Angle equal to an Angle given The Position Let A B. be a Line bounded upon that which one would make a Portion of a Circle capable of containing an Angle equal to the Angle given C. The Practice Make the Angle B A D. equal to the Angle C. Page 16. Page 2. Page 2. Elevate upon A D. the Perpendicular A E. Divide the Line A B. into two equally in H. Elevate the Perpendicular H E. from the Section F. and from the Interval F A. Describe the Portion of the Circle A E B. All the Angles that you shall make within the Portion of a Circle and upon the Line given A B shall be all equal to the Angle C. PROPOSITION IX To find the Center of a Circle given The Position Let A B C. be a Circle propounded whereof we must find the Center The Practice Draw at discretion the right Line A B. bounding it self at the Circumference B C. Cut this right Line A B. into two by the Line D C Cut also this right Line C D. into two equally in F. The point F. shall be the Center demanded of the Circle A B C. Here endeth the Seven of Harts See the Eight of Harts PROPOSITION X. To finish a Circumference begun whereof the Center is lost The Position Let A B C. be the part of the Circumference given we must find the Center that we may finish it The Practice Place at discretion the three Points A B C. within the circumference begun From the Points A and B. make the Sections E and F. Draw the right Line E F. from the Points B and C. Make the Sections G And H. Draw the right Line G H. From the Intersection and Center I. and from the Interval I A. finish the Circumference began PROPOSITION XI To describe a Circumference by three Points given The Position Let A B C. be the three Points by the which one would pass a Circumference The Practice From the Points given A B C. describe the three Circles D E H. D E F. F G L. of the same Interval dividing each other in the points D and E F and G. Draw the right Lines D E F G. until that they meet each other in I. From the point I. and the Interval I A. describe the Circumference required This Practice is like the former Here endeth the Eight of Harts See the Nine of Harts PROPOSITION XII To describe an Oval upon a Length given The Position Let A B. be the Length upon the which we must frame an Oval The Practice Divide the length given A B. into three parts equal A C D B. Page 18. from the points C and D. and from the Interval C A. Describe the Circles A E F B E F. from the Sections E and F. and from the Interval of the Diameter E H. Describe the Arches I H O P. A I H B P O. shall be the Oval required PROPOSITION XIII To describe an Oval upon two Diameters given The Position A B C D. are the Diameters upon the which we must frame an Oval The Practice Make the Rule M O. equal to the great half Diameter A E. upon the which you shall make the length M N. equal to the half Diameter C E. This Rule being so ordered Place it so upon the Demands A B. C D. That the point N. sliding upon the line A B. the end O. may never leave the line C D. running on so the said Rule M O. Describe the Oval by the end M. PROPOSITION XIV To find the Center and the two Diameters of an Oval The Position Let A B C D. be the Oval propounded whereof we must find the Centers and the Diameters The Practice Within the Oval propounded A B C D. draw at discretion Pgae 10. the Lines parallels A N. H I. Cut these Lines A N. H I. into two equally in L and M. Draw the Line P L M O. Cut it into two equally in E. and the point E. shall be already the Center from the Point E. Describe at discretion the Circle F H Q. cutting the Oval in F and G. from the Sections F and G. Draw the right Line F G. Cut it into two equally in R. Draw the great Diameter B D. by the Points E R. from the Center E. Page 10. Draw the less Diameter A E C. parallel to the Line F G. This is it which was propounded Here endeth the Nine of Harts See the Ten of Hearts PROPOSITION XV. To frame a Figure Rectilinear upon a right line bounded like unto a Figure Rectilinear Propounded Let A B. be the line upon the which we must frame a Figure like to the Figure C D E F. The Practice Draw the Diagonal C E. make the Angle B A G. equal to the Angle F C E. make the Angle A B G. equal to the Angle C F E. the triangle A B G. shall be like to the Triangle C F E. The same Make the Triangle A G H. like to the Triangle C E D. the whole Figure A B G H. shall be like to the whole Figure C D E F. PROPOSITION XVI Upon a right line Propounded to frame two Rectangles according to one Reason given A B is the line upon which we must frame two Rectangles which may be between them according to the reason of C to D. The Practice Cut the line A B. at the point E according to the reason or proportion of C to D. make the square A B H F. draw the line E I. parallel to the line A F. B E I H A E I F shall be the Rectangle required the Rectangle A I. is to the Rectangle E. H. as the line D. is to the line C. See the Ten of Hearts PROPOSITION I. Within a Circle given to inscribe a Triangle Equilateral an Exagone and a Dodecagone Let A B G. be the Circle within which we must inscribe a Triangle Equilateral c. The Practice Of the Triangle Equilateral From the point as A. and from the interval of the half Diameter A B. describe the Arch C B D. draw the right line D C. bring this Interval C D. from the
E F G. Here endeth the Six of Spades See the Seven of Spades PROPOSITION X. About a Square to circumscribe a Pentagone Let A B C D be the Square about the which we must circumscribe a Pentagone Prolong the side C B. towards N. divide the side A B. into two equally in R. Elevate the Perpendicular R V. from the points B D C. and from the same Interval B R. divide the Arches R N S T S T. divide the Arch R N. into five equal parts R H G F E N. make the Angle R B V. from the opening of two parts R G. make the Angles S C T S D T. from the opening of one part R H. prolong the lines V B C T in O. make the line O Q. equal to the line O V. draw the other sides on the sume manner and you shall have the demanded PROPOSITION I. To find a line which may be a mean proportional between two others LEt A and B be the lines between the which we must find a third which may be proportional to them The Practice Draw a line Undeterminate G H. make C E. equal to the line A. make E D. equal to the line B. divide C D. into two equally in I. from the point I. and from the Interval I C. describe the Demy-circle C F D. elevate the Perpendicular E F. This line E F. shall be the mean proportional between A and B. according as it is propounded Here endeth the Seven of Spades See the Eighth of Spades PROPOSITION II. There being given the sum of the Extreams and the mean proportional to discern the Ends and Extreams Let A B be the sum of the Ends that is to say two Grandeurs or Greatnesses the one at the end of the other without distinction whereof the line C is the mean proportional and by the means of which we must find the point where the Extreams or Ends do joyn themselves The Practice Divide the sum or the line A B. into two equally in G. from the point G. and from the Interval G A. describe the Demy-circle A E B. Elevate the perpendicular B D. equal to the mean C. draw the line D E parallel to the line A B. From the section E. draw the line E F. parallel to the line B D F shall be the point where the Extreams join themselves and so C or his equal E F. shall be the mean between the Extreams A F and F B. PROPOSITION III. There being given the mean of three Proportionals and the difference of the Extreams or Ends to find the Extreams Let G H be the mean Proportional and A B the difference of the Extreams we must find the length of the Extreams The Practice Elevate the perpendicular B C. at the end of the difference A B. and equal to the mean G H. divide the difference A B. into two equally in D. prolong it towards E and F. from the point D. and from the interval D C. describe the Demy-circle E C F. B E B F shall be the Extreams demanded Here endeth the Eight of Spades See the Nine of Spades PROPOSITION IV. From a right line given to cut off a part which may be a mean proportional between the rest and another right line Propounded Let A A be the line from the which we must cut off a part which may be the mean proportional between the part that shall remain and the line proposed B B. The Practice Draw the line Indeterminate C D. divide the lines D E E C. equal to the lines A A and B B. describe the Demy-circle C F D. elevate the perpendicular E F. divide the line C E. into two equally in B. from the point B. and from the interval B F. describe the Arch F G. divide the part demanded A G. equal to the part E G. A H shall be the mean proportional between the rest H I. and the other line propounded B B. PROPOSITION V. There being given two right lines to find a Third Proportional A B A C are the two right lines given we must find a third which may be proportional to them The Practice Make at discretion the Angle D N E. divide the part N H. equal to the line A B. divide the part N O. equal to the line A C. divide also H D. equal to the line A C. bring the line H O. draw the line D E. parallel to the line H O. E O shall be third proportional demanded Here endeth the Nine of Spades See the Ten of Spades PROPOSITION VI. To find a fourth Proportional A B C are the three lines proposed we must find a fourth which may be to the third as the second to the first The Practice Make at discretion the Angle G D H. divide the part D E. equal to the line A. divide the part D F equal to the line B. divide the part E G. equal to the line C. bring the line E F. draw the line G H. parallel to the line E F. F H shall be fourth proportional demanded PROPOSITION VII Between two right lines given to find two means proportional Let I and H be the lines propounded between the which we must find two mean proportional The Practice Draw the line A B. equal to the line H. let down the Perpendiculars B C. equal to the line I. bring the line A C. divide this line A C. into two equally in F. elevate the Perpendiculars A O C R. From the point or center F. describe the Arch D E. in such manner that the Cord D E. touch the Angle B. A D C E shall be the means proportional between the lines given I and H. Here endeth the Ten of Spades PROPOSITION VIII To divide two right lines given each into two parts so that the four Segments may be proportional A B A C are the lines propounded to be The Practice Make the right Angle B O C. divide the line B O. equal to the line A B. divide the line O C. equal to the line A C. bring the subtendent B C. describe the Demy-circle B D O. from the section D. bring the line D E. parallel to the line C O. the line D F. parallel to the line E O. A B shall be divided in E. O C shall be divided in F. So that B E shall be to E D as E D is to D F and E D to D F as D F is to F C. Here endeth the King of Spades See the Queen of Spades PROPOSITION IX There being given the Excess of the Diagonal of a square about the side to find the greatness of the said side Let A B be the Excess of the Diagonal of a square above its side whereof we must find the greatness The Practice Elevate the perpendicular B C. equal to the Excess B A. draw the line A C. prolonged towards D. from the point C. and from the interval C B. describe the Arch B D. A D shall be the side of the square
whereof A B is the Excess of the Diagonal A E above the said side A D. PROPOSITION X. To divide a right line Terminated within the mean and extream reason Let A B be the line which we must divide in such manner as the Rectangle composed of the whole line and of one of the two parts may be equal to the square framed upon the other part The Practice Elevate the perpendicular A D. prolong it towards D. make A C. equal to the half of A B. from the point C. and from the Interval C B. describe the Arch B D. from the point A. and from the interval A D. describe the Arch E. The line A B. shall be divided in E. according to the proposition for if you make the Rectangle A H of the whole A B and of the part B E it shall be equall to the square A F. framed upon the other part A E. Here endeth the Queen of Spades PROPOSITION XI To divide a right line terminated according to the reasons given Let A B be the line propounded to be divided according to the Reasons C.D.E.F. The Practice From the point or end A. draw at discretion the line A G. make A H. equal to the line or reason C. make H I. equal to the line D. make I L. equal to the line E. make L M. equal to the line F. draw the line B M. bring the lines L N I O H P. parallels to the line B M. the line A B shall be divided in the points P O N according as it is demanded Here endeth the Knave of Spades Mechanick Powers OR A MANUSCRIPT OF Monsi Des-Cartes See the Ace of Clubs The Explication Of Engines by help of which we may raise a very great Weight with small Strength THE invention of all these Engins depends upon one sole Principle which is that the same Force that can lift up a Weight for Example of 100 l. to the height of one Foot can lift up one of 200 l. to the height of half a Foot or one of 400 l. to the height of a fourth part of a Foot and so of the rest be there never so much applied to it and this Principle cannot be denied if we consider that the Effect ought to be proportioned to the Action that is necessary for the Production of it So that if it be necessary to employ an Action by which we may raise a Weight of 100 l. to the height of two foot for to raise one such to the height of one foot only this same ought to weight 200 l. for it 's the same thing to raise 100 l. to the height of one foot and again yet another 100 l. to the height of one foot as to raise one of 200 l. to the height of one foot and the same also as to raise 100 l. to the height of two feet Now the Engines which serve to make this Application of a Force which acteth at a great space upon a Weight which it causeth to be raised by a lesser are the Pulley the inclined Plane the Wedge the Capsten or Wheel the Screw the Lever and some others for if we will not apply or compare them one to another we cannot well number more and if we will apply them we need not instance in so many The Pulley Trochlea Let A B C be a Chord put about the Pulley D to which let the Weight E be fastned and first supposing that two Men sustain or pull up equally each of them one of the ends of the said Chord it is manifest that if the Weight weigheth 200 l. each of those Men shall employ but the half thereof that is to say the force that is requisite for sustaining or raising of 100 l. for each of them shall bear but the half of it Afterwards let us suppose that A one of the ends of this Chord being made fast to some Nail the other C be again sustained by a Man and it is manifest that this Man in C needs not no more than before for the sustaining the Weight E more force than is requisite for the sustaining of 100 l. because the Nail at A doth the same Office as the Man which we supposed there before in fine let us suppose that this Man in C do pull the Chord to make the Weight E to rise and it is manifest that if he there employeth the force which is requisite for the raising of 100 l. to the height of two foot he shall raise this Weight E of 200 l. to the height of one foot for the Chord A B C being doubled as it is it must be pulled two feet by the end C to make the Weight E rise as much as if two men did draw it the one by the end A and the other by the end C each of them the length of one foot only There 's always one thing that hinders the exactness of the Calculation that is the ponperosity of the Chord or Pulley and the difficulty that we meet with in making the Chord to slip and in bearing it But this is very small in comparison of that which raiseth it and cannot be estimated save within a small matter Moreover its necessary to observe that it s nothing but the redoubling of the Chord and not the Pulley that causeth this force for if we fasten yet another Pulley towards A about which we pass the Chord A B C H there will be required no less force to draw H towards K and so to lift up the Weight E than there was before to draw C towards G. But if to these two Pulleys we add yet another towards D to which we fasten the Weight and in which we make the Chord to run or slip just as we did in the first then we shall need no more force to lift up this Weight of 200 l. than to lift up 50 l. without the Pulley because that in drawing four foot of Chord we lift it up but one foot and so in multiplying of the Pulleys one may raise the greatest Weights with the least Forces It s requisite also to observe that a little more Force is always necessary for the raising of a Weight than for the sustaining of it which is the reason why I have spoken here distinctly of the one and the other The inclined Plane If not having more Force than sufficeth to raise 100 l. one would nevertheless raise this body F that weigheth 200 l. to the height of the line B A there needs no more but to draw or rowl it along the inclined Plane C A which I suppose to be twice as long as the line A B for by this means for to make it arrive at the point A we must there employ the Force that is necessary for the raising 100 l. twice as high and the more inclined this Plane shall be made so much the less Force shall there need to raise the Weight F. But yet there is to be rebated from this
it is near to C. of which the reason is that the Weights do there mount less as it is easie to understand if having supposed that the line C O H is parallel to the Horizon and that A O F cutteth it at right Angles we take the point G equidistant from the point F and H and the point B equidistant from A and C. and that having drawn G S perpendicular to F O we observe that the line F S which sheweth how much the Weight mounteth in the time that the force operates along the line A B is much lesser than the line S O which sheweth how much it mounteth in the time that the force operates along the line B C. And to measure exactly what his force ought to be in each point of the curved line A B C D E it is requisite to know that it operates there just in the manner as if it drew the Weight along a plane circularly inclined and that the inclination of each of the points of this circular plane were to be measured by that of the right line that toucheth the circle in this point As for Example when the force is at the point B for to find the proportion that it ought to have with the Ponderosity of the Weight which is at that time at the point G its necessary to draw the Tangent line G M and to account that the ponderosity of the Weight is to the force which is required to draw it along this plane and consequently to raise it according to the circle F G H as the line G M is to S M. Again for as much as B O is triple of O G the force in B needs to be to the Weight in G but as the â…“ of the line S M is to the whole line G M. In the self same manner when the force is at the point D to know how much the Weight weigheth at I its necessary to draw the Contingent line betwixt I and P and the right line I N perpendicular to the Horizon and from the point P taken at discression in the line I P provided that it be below the point I you must draw P N parallel to the same Horizon to the end you may have the proportion that is betwixt the line I P and the â…“ of I N for that which is betwixt the ponderosity of the Weight and the force that ought to be at the point D for the moving of it and so of others Where nevertheless you must except the point H at which the contingent line being perpendicular to the Horizon the weight can be no other than triple the force which ought to be in C for the moving of it in the points F and K at which the contingent line being parallel to the Horizon it self the least force that one can assign is sufficient to move the Weight Moreover that you may be perfectly exact you must observe that the lines S M and P N ought to be parts of a circle that have for their center that of the Earth and G M and I P part of Spirals drawn between two such Circles and lastly that the right lines S M and I N both tending towards the center of the Earth are not exactly parallels And furthermore that the point H where I suppose the contingent line to be perpendicular to the Horizon ought to be some small matter nearer to the point F than to K at the which F and K the contingent lines are parallels to the said Horizon This done we may easily resolve all the difficulties of the Balance and shew that then when its most exact and for instance supposing its center at O by which it is sustained to be no more but an indivisible point like as I have supposed here for the Leaver if the arms be declined one way or the other that which shall be the lowermost ought evermore to be adjudged the heavier so that the center of Gravity is not fixt and immovable in each several body as the Ancients have supposed which no Person that I know of hath hitherto observed But these last considerations are of no moment in Practice and it would be good for those who set themselves to invent new Machines that they knew nothing more of this business than this little which I have now writen thereof for then they would not be in danger of deceiving themselves in their computation as they frequently do in supposing other Principles A LETTER OF Monsi Des-Cartes To the Reverend Father Marin Mersenne See the Deux of Clubs Reverend Father I Did think to have deferred Writing to you yet 8 or 15 days to the end I might not trouble you too often with my Letters but I have received yours of the First of September which giveth me to understand that its an hard matter to admit the principle which I have supposed in my Examination of the Geostatick Question and in regard that if it be not true all the rest that I have inferred from it would be yet less true I would not only to day defer sending you a more particular Explication It s requisite above all things to consider that I did speak of the Force that serveth to raise a Weight to some height the which Force has overmore two Dimensions and not of that which serveth in each Point to sustain it which hath never more than one Dimension insomuch that these two Forces differ as much the one from the other as a Superficies differs from a Line for the same Force which a Nail ought to have for the susstaining of a Weight of 100 l. one moment of time doth also suffice for to sustain it the space of a Year provided that it do not diminish but the same Quantity of this Force which serveth to raise the Weight to the height of one foot sufficeth not eadem numero to raise it two feet and it s not more manifest that 2 and 2 makes 4 than its manifest that we are to employ double as much therein Now for as much as that this is nothing but the same thing that I have supposed for a Principle I cannot guess on what the Scruple should be grounded that Men make of receiving it but I shall in this place speak of all such as I suspect which for the most part arise only from this that Men are beforehand overknowing in the Mechanicks that is to say that they are pre-occupied with Principles that others prove touching these matters which not being absolutely true they deceive the more the more true they seem to be The first thing wherewith a Man may be pre-occupied in this business is That they many times confound the consideration of Spaces with that of Time or of the Velocity so that for Example in the Leaver or which is the same the Ballance A B C D having supposed that the Arm A B is double to B C and the Weight in C double to the Weight in A and also that
THE USE Of the GEOMETRICAL Playing-Cards As also a Discourse of the Mechanick Powers By Monsi Des-Cartes Translated from his own Manuscript Copy SHEWING What Great Things may be performed by MECHANICK ENGINES in removing and raising Bodies of vast Weights with little Strength or Force LONDON Printed and Sold by J. Moxon at the Atlas in Warwick-Lane 1697. See the Ace of Diamonds The Definition of a POINT THE Point is that which hath not any Part. By this Definition it is easie to conceive that the Point hath neither length nor breadth nor depth and that also it is not sensible but only Intellectual seeing that there is nothing which falleth under sense which hath not a quantity and that there is no quantity without Parts which would altogether contradict this Definition Nevertheless as none can make any Operation but by the Interposition of Corporal things they represent therefore the Mathematical Point by the Point Physical which is the Object of the sight the smallest and least sensible which hath no Geometrical greatness divisible to our sense and is made with the Point of a Needle or with the end of the Point of the Compass of a Pen or Pensil as the Point noted by A. The Point Central or Center is a Point by which a Circle is drawn or a Circumference or rather it is the midst of a Figure as C. The Point Secant is a Point where the Lines do interdivide themselves and which is ordinarily called a Section as C D. The Definition of a LINE THe Line is a Length without Breadth The Line is no other thing then the passing of the Point from one place to another and it would be unperceivable if one should not set it forth by a Point Physical the which by its flowing doth represent to us as A B C D E F. There are many sorts of Lines as the Point which is the Original thereof is able to receive different Motions nevertheless they take into consideration only two Single and Principal ones the Straight and the Crooked and a third also which they call the Mixt because that it is composed of the two first The right or straight Line is that which is equally comprized within its extremities Otherwise it is that which floweth from one Point to another without any turnings aside as A B. The Crooked Line is that which turneth or wandreth from its Extremities by one or more Turnings aside as C D. When as this Line is described with a Compass they call it Circular as E. The Mixt Line is that which is Straight and Crooked as the Line V. The LINE is distinguished into Finite and Infinite into Apparent and Occult or Hidden THe Line Finite is a Line Bounded which containeth or suppofeth a length necessary as A. The Infinite is a Line undetermined which hath no precise length as B. The Apparent or Traced out is that which is described with Ink or Pencil as A B. The Occult or White which is drawn only with the Point of the Compass or marked with Points and therefore they call it the Pointed or Line with Points as C. Here endeth the Ace of Diamonds See the Duce of Diamonds The Line receiveth also divers Denominations according to its divers Positions and Proprieties A Perpendicular is a right Line which falleth or lifteth it self up upon another maketh the Angles of the one part and the other equal between themselves as A B. A Plumb Line is that which goeth from high to low without bending either to the right or to the left and which would pass through the Center of the World if it were prolonged infinitely as C. A Line Horizontal is a Line in an equal poize which inclineth it self equally on the one part and the other as D E. Lines Parallels are those which follow each other by an equal distance as H. An Oblique Line which is neither Horizontal nor a Plumb-line but of a Bias as F G. The Basis is the Line upon which the Figure reposeth it self as I L. Sides are those Lines which encompass a Figure as I N. N M. THe Diagonal is a right Line that traverseth a Figure and that which endeth at two Angles opposite as A B. The Diameter is a right Line which traverseth or passeth through a Circular Figure by its Center and which ends at the Circumference A Spiral Line is a Crooked Line which parteth from its Center and groweth farther off in proportion as it turneth about as E E. The Cord or Subtendant is a right Line which is join'd to an Arch or Bow by its ends as G H. The Arch or Bow is a Part of a Circumference as G I H. A Line Tangent is that which toucheth any Figure without dividing it and without being able to divide or pass through it although it were prolonged as L M. A Line Secant which Crosseth which divideth or traverseth as L O. M O. See the Tray of Diamonds The Definition of the ANGLE AN Angle is the indirect meeting of two Lines at one and the same Point or rather it is the space encompassed between the indirect meeting or concourse of two lines joining together in one Point as A B C. When as this Concourse is made of two right Lines the Angle is called Rectilinear and when it is made of two crooked Lines it is called Curvi-linear but when it is made of one right Line and one crooked Line it is called Mixtilinear A. The Angle Rectilinear B. The Angle Curti-linear C. The Angle Mixt-linear or Composite The Angle Rectilinear according as it is more or less open it receiveth particular Denominations as Right Sharp Obtuse or Blunt So that these Terms of Rectilinear Curvi-linear and Mixt are in respect of the Quality of the Lines and these of Right Sharp and Obtuse are in respect of the quantity of the space inclosed within the said Lines It is a right Angle when one of the Lines is Perpendicular upon the other as E D F. The Angle is sharp when as it is less open than the right Angle as E D G. The Angle is Obtuse when it is more open than the Right Angle as F D G. The Letter D in the midst sheweth the Angle Here endeth the Tray of Diamonds See the Four of Diamonds The Definition of the Superficies THe Superficies is that which hath length and breadth without depth According to the Geometricians the Superficies is a Production of the flowing forth of the Line as the Line is a Production of the Point And thus we must conceive that the Line E F. flowing towards G H. doth make the Superficies E F G H. which is an extension bounded with Lines which hath nothing but length and breadth without any depth or thickness which is called the Surface or Figure if one consider it in respect of its extremities which are the Lines that enclose it If the Superficies be on the upper part it is called a Convex if it be in the inner or
Demicircle A D B. cutting the Circle in D. from the point given A. Draw the right Line A E. by the point D. This right Line A E. shall be the Line touching the required Here endeth the Ace of Harts See the Duce of Harts PROPOSITION XI To draw a right Line which toucheth a Circle at a Point propounded The Position Let A B C. be the Circle given within the Circumference of which is the Point A. propounded The Practice From the Point or Center D. draw the Line D F. by the Point propounded A. at the Point propounded A. and upon the Line D F. Draw the Perpendicular A H. prolonged towards I. This Line Tangent H I. shall touch the Circle at the Point propounded A. the which is required by the Proposition PROPOSITION XII A Circle being given and a straight Line that toucheth it to find the Point where it toucheth The Position Let A B C. be the Circle touched by the Line G H. we must find the Point where it toucheth The Practice From the Center of the Circle F. let down the Perpendicular F C. upon the Line touching D E. The Section C. shall be the Point of Touching demanded PROPOSITION XIII To describe a Line Spiral upon a straight Line given The Position Let I L. be the Line upon which we would describe a Line Spiral The Practice Provide the half of the Line I L. into as many equal Parts Page 18. as you would describe the Revolution Example If you would divide it into Four Divide the half B I. into four equal parts B C E G I. Divide also B C. into two equally at A. Page 12. From the Point A. draw the Demi-circles B C. D E. F G. H I. From the point B. draw the Demicircles C D. E F. G H. I L. and you shall have the Spiral Line required Here endeth the Duce of Harts PROPOSITION XIV Between the Points given to find two other directly interposed The Position Let A and B. be the Points given between the which we must find two other Points directly interposed by the means whereof we may draw a straight Line from the Point A. to the point B. with a short Ruler The Practice From the Points A and B. make the Sections C and D. From the Points C and D. make the Sections G and H. The Points G and H. shall be the demanded by the means of which one may draw three ways a right Line from the point A. to the point B. the which could not be done in one with a Ruler which should be shorter than the space between A and A. The Second Part of the Construction of Plain Figures PROPOSITION I. To frame a Triangle Equilateral upon a right Line given and bounded The Position Let A B. be the Line given upon the which we must frame a Triangle Equilateral The Practice From the end A. and the Interval A B. describe the Arch B D. from the end B. and the See the Tray of Harts Interval B A. Describe the Arch A E. from the Section C. Draw the Lines C A. C B. A B C shall be the Triangle Equilateral demanded PROPOSITION II. To make a Triangle of three straight Lines equal to three straight Lines given The Position Let A B C be the three Lines given we must make a Triangle of three right Lines equal to them The Practice Draw the right Line D E. equal to the Line A A. from the Point D. and from the Interval B B. Describe the Arch G F. from the point F. and the Interval C C. Describe the Arch H I. from the Section O. Draw the Lines O E O D. The Triangle D E O. shall be comprised of three right Lines equal to the three right Lines given A A B B C C. Here endeth the Tray of Harts See the Four of Harts PROPOSITION III. To frame a Square upon one right Line given and bounded The Position Let A B be the right Line given and bounded upon the which we must frame a Square The Practice Elevate the Perpendicular A C. from the point A. Page 4. Describe the Arch B C. from the Points B and C. and from the Interval A B. Make the Section D. from the point D. Draw the Lines D C D B. A B C D shall be the Square demanded framed upon the right Line given A B. PROPOSITION IV. To frame a Pentagone Regular upon a right Line given The Position Let A B be the Line given upon the which we must frame a Pentagone The Practice From the end A. and from the Interval A B. Describe the Arch B D F. Elevate the Perpendicular A C. Divide the Arch B C. into five equal Parts I D L M. Draw the right line A D. Cut the Bases A B. into two equally in O. Elevate the Perpendicular O E. from the Section E. and from the Interval E A. Describe the Circle A B F G H. Bring five times the Line A B. within the circumference of the Circle and you shall have a Pentagone Regular Equiangle Equilateral A B F G H. PROPOSITION V. To frame an Exagone Regular upon a right Line given The Position Let A B. be the right Line upon the which we must frame an Exagone The Practice From the end A and B. and from the Interval A B. Describe the Arches A C B C. from the Section C. Describe the Circle A B E F G. Bring Six times the Line given A B within the circumference and you shall have an Exagone Regular A B E F G D framed upon the Line given A B. Here endeth the Four of Harts See the Five of Harts PROPOSITION VI. Upon a right Line given to describe such a Poligone as you would have from the Exagone unto the Dodecagone The Position Let A B be the Line upon the which that must frame an Exagone or an Eptagone or an Octogone c. The Practice Cut the Line A B. into two equally in O. Page 10. Elevate the Perpendicular O I. From the Point B. describe the Arch A C. Divide A C. into six equal parts M N. P Q R. this may make an Eptagone if you will From the Point C. and the Interval of one part C M. describe the Arch M D. D shall be the Center to describe a Circle capable of containing seven times the Line A B. If you would make an Octogone From the point C. and the Interval of two parts C N. describe the Arch N E. E. shall be the Center to describe a Circle capable of containing eight times the Line A B If you would make an Enneagone You must take the three parts C P. And so likewise of others always augmenting it by one part Here endeth the Five of Harts See the Six of Harts PROPOSITION VII Upon a right Line given to frame such a Poligone as one would have from 12 to 24 sides The Position Let A B. be the Line upon the which
that it may be they have never been well demonstrated by any Man but neither have I made use of them save only with a design to shew that this Principle extends it self to all matters of which one treateth in the Staticks or rather I have made use of this occasion for to insert them into my Treatise for that I conceived that it would have been too dry and barren if I had therein spoken of nothing else but of this Question that is of no use as of that of the Geostaticks which I purposed to Examine Now one may perceive by what hath already been said how the Forces of the Leaver and Pulley are demonstrated by my principle so well that there only remains the inclined plane of which you shall clearly see the demonstration by this Figure in which G F represents the first Dimension of the Force that the Rectangle F H describeth whilst it draweth the Weight D along the plane B A by the means of a Chord parallel to this plane and passing about the Pulley E in such sort that H G that is the height of this Rectangle is equal to B A along which the Weight D is to move whilst it mounteth to the height of the line C A. And N O represents the first Dimension of such another Force that is described by the Rectangle N P in the time that it is raising the Weight L to M. And I suppose that L M = B A = 2C A and that N O F G⸬O P-G H. This done I consider that at such time as the Weight D is moved from B towards A one may imagine its motion to be composed of two others of which the one carrieth it from B R towards C A to which operation there 's no Force required as all those suppose who treat of the Mechanicks and the other raiseth it from B C towards R A for which alone the Force is required in so much that it needs neither more nor less Force to move it along the inclined plane B A than along the perpendicular C A. For I suppose that the unevenness c. of the Plane do not at all hinder it like as it is always supposed in treating of this matter So then the whole Force F H is employed only about the raising of D to the height of C A and for as much as it is exactly equal to the Force N P that is required for the raising of L to the height of L M double to C A I conclude by my principle that the Weight D is double to the Weight L. For in regard that it is necessary to employ as much Force for the one as for the other there is as much to be raised in the one as in the other and no more knowledge is required than to count unto two for the knowing that it is alike facile to raise 200 l. from C to A as to raise 100 l. from L to M since that L M = 2C A. You tell me moreover that I ought more particularly to explain the nature of the Spiral Line that representeth the plane equally enclined which hath many qualities that render it sufficiently knowable For if A be the Center of the Earth and A N B C D the Spiral Line having drawn the Right Lines A B A D and the like there 's the same proportion betwixt the Curved Line A N B and the Right Line A B as is betwixt the Curved Line A N B C and the Right Line A C or betwixt A N B C D and A D and so of the rest And if one draw the Tangents D E C F and B G the Angles A D E A C F A B G c. shall be equal As for the rest I will c. Reverend Father Your very humble Servant Des-Chartes A LETTER OF Monsi de Robberal TO Monsi de Fermates COUNSELLOR OF THOULOUSE CONTAINING Certain Propositions in the Mechanicks Monsieur I Have according to my promise sent you the Demonstration of the fundamental proposition of our Mechanicks in which I follow the common method of explaining in the first place the Definitions and Principles of which we make use We in general call that quality a Force or Power by means of which any thing whatever doth tend or aspire into another place than that in which it is be it downwards upwards or side-ways whether this Quality naturally belongeth to the Body or be communicated to it from without From which definition it followeth that all Weights are a Species of Force in regard that it is a Quality by means whereof Bodies do tend downwards We often also assign the name of Force to that very thing to which the Force belongeth as a ponderous Body is called Weight but with this pre-caution that this is in reference to the true Force the which augmenting or diminishing shall be called a greater or lesser Force albeit that the thing to which it belongeth do remain always the same If a Force be suspended or fastened to a flexible Line that is without Gravity and that is fastened at one end to some Fulciment or Stay in such sort as that it sustain the Force drawing without impediment by this line the Force and the line shall take some certain position in which they shall rest and the line shall of necessity be straight let the line be termed the Pendant or line of Direction of the Force and let the point by which it s fastned to the Fulciment be called the point of Suspension which may sometimes be the Arm of a Leaver or Ballance and then let the line drawn from the Center of the Fulciment of the Leaver or Ballance to the point of suspension be named the Distance or the Arm of the Force which we suppose to be a line fixed and considered without Gravity Moreover let the Ang. comprehended betwixt the Arm of the Force and the line of Direction be termed the Ang. of the Direction of the Force Axiom 1. After these Definitions we lay down for a Principle that in the Leaver and in the Ballance equal Forces drawing by Arms that are equal and at equal Angles of Direction to draw equally And if in this Position they draw one against the other they shall make an Equilibrium but if they draw together or towards the same part the Effect shall be double If the Forces being equal and the Angles of Direction also equal the Arms be unequal the Force that shall be suspended at the greater Arm shall work the greater Effect As in this Figure the Center of the Ballance or Leaver being A if the Arms A B and A C are equal as also the Angles A B D and A C E the equal Forces D and E shall draw equally and make an Equilibrium So likewise the Arm A F being equal to A B the Ang. A F G to the Ang. A B D and the Force G to D these two Forces * In M. S. Coppy it s C and D. G