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
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
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.
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
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
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