with great facility either heighth breadth or depth as followeth PROPOSITION I. How to take a height accessible LEt it be proposed to take the heighth of the Tower AB to the base or foot whereof one may easilie approach dispose the crosse of your instrument in such sort that DCE be of equall distance there one from the other as thus settle your crosse vpon the first point of the index then setting the instrument to your eye goe either neerer or retire farâ—her from the obiect you desire to measure vntill you see A the highest part thereof by the two extremities of the crosse CD the inden being paralled to the earth which will happen in the point C and not else-where that being measured will be alwayes the iust height of the Tower required onelie obserue that you must alwayes adde to the distance betweene you and the Tower the length of the foot which supporteth your instrument and so you shall haue preciselie the heighth required for scarce euer will it happen that the instrument may be placed leuell with the base or foote oâ— the height required but if that should chance to be then the extremitie of the index C would ariue in the point G and so nothing to be added but onelie measure the distance betweene G and B and that would be the iust height required but if it chance that you may not plant the instrument in the point C by reason of some inconueniences which may happen retire further backe and put forward the crosse to the second point of the index and then accord your visuele lines to see the point A the index alwaies being paralel to the ground which then will happen twice the height required distant from the foot thereof but if it happen that going backe you ascend anie little mountaine or descend into any little valley you are then to obserue some point in the wall of the Tower which the index shall direct you to by the line visuel adde onlie the height thereof vnto halfe the distance twixt you and the Tower and that shall be the height required most exactlie and thus you shall neuer faile in your operations PROPOSITION II. How to take a height inaccessible or one height vpon another height SVppose the altitude BC to be required to the foot whereof one may come may come no neerer then to the point D then at the point D accord your visuell lines in B and C the index being alwaies leuell with the ground and the crosse fixed vpon the first marks or point of the index then set a marke there in D and going backe towards F as farre as E and put forward the crosse to the second point of the index and then direct your lines visuel againe in B and in C which will arriue in E and not elsewhere then measure the distance betweene DE which will be the altitude precisely required but if you desire the heighth of the Tower B and mountaine leuell with the foote of your instrument then you must adde the length of the staffe which supporteth it as in former obseruations And to haue the altitude A B vpon the top of BC you must take two other obseruations in FG as before setting your instrument in F the crosse being fixed vpon the second point of the index and direct your visuell lines in A and in C and then set vp a mark in F and goe backe putting forward the crosse to the third point of the index and direct your visuell lines againe in A and in C which will happen in G and not elsewhere then measure the distance betweene FG which shall bee equall to the altitude of the Tower AC and haâ—â—ing taken already the altitude BC you may verie easilie substract the little altitude BC from the greater Altitude AC and then will rest the altitude AB which is required PROP. III. How to take any distance vpon a place accessible or inaccessible IF it were proposed to take the distance AB and that the place were accessible onelie in the middle vpon the â—â—ne CF then dispose your instrument as before the crosse placed vpon the first point of the index and going forward or backward vpon â—he line CF direct your visuel lines 〈◊〉 A and in B by the two extreamiâ—es of the crosse the end of the index resting against your eie then measure the distance betweene you and the point F and that shall be iust half the distance required betweene AB but if the place be inaccessible so that you may not approch neerer then the point D put forward the crosse to the second point of the index then direct your visuel lines to AB and leaue a marke at D and put forward the crosse one point more and going backe vpon the line DE vntill you may direct your visuel lines by the extreamities of the crosse againe in A and B then measure the distance between DE and that will be halfe the distance betweene AB and so may you operate going backe and putting forward the crosse PROPOSITION IV. Another manner how to take a distance inaccessible SVppose the distance AB to bee taken and that B is the neerest place that may be required then there must be set a marke and withdrawing backward in a straight line towards C and there againe plant another marke at C then going right towards one side as towards F counting your places equall to the number which you haue alreadie found B and C and there direct your visuel lines in BGC as in this figure at D leauing there a mark goe straight along towards F not moouing the crosse of your instrument and going along make trial where your visuel lines may be directed again in A and in C which will be in the point E and not else-where then leaue there a marke and measure the distance betweene DE and that shall bee the distance which you require which is AB the demonstration of this proposition is grounded vpon the second and fourth propositions of 6 of Euclide PROP. V. How to take a distance onely vpon a line parallell to it LEt the distance AB bee required the which may neither bee seene nor come neere vnto but onelie vpon the line or banke CK then vpon that banke draw out a straight line with markes as the line CK paralell to the wall AB and set a marke in K then goe backe towards D and direct your visuel lines in B and in K by the extreamities of the crosse GH then leauing the crosse in the same state and a marke at D withdraw your selfe further back towards C vntill you may direct your visuel lines to A and K which will happen in C and not elsewhere and leaue there another marke then measure the distance betweene CD and the same shall bee the breach AB equall to CD and the demonstration of this proposition is grounded in the 29. and 33. propositions of the 1. of Euclide PROP. VIII How to
first describe the Morter-peece and the vse thereof that being an instrument the most noble the most vniuersal the best of greatest effect and of most wonderfull operation of all the instruments the practise wherof may be vsed amongst Fire-workes for warres for this instrument may serue for a Petard to split breake and hurle down dores gates or walles likewise to massacre teare in pieces ouerthrow and confound assailants of any place or breach and for diuers other most worthy offices and accidentall occasions which happen in the troubles of warres whereof I will not here make a long weary and tedious discourse but in few words cleerely set downe the vse of this instrument for the shooting of diuers sorts of granads stones or other weighty burthens to ruine rebels their habitations and dwelling places Then first of all I will treat of the mettle whereof this instrument ought to be made and measures appertaining thereunto CHAP. II. The manner how to make the Morter-Peece THis Instrument may bee made of diuers sorts of mettle or stuffe according to the means times and occasions which shall offer themselues to those who would make them or cause them to bee made The first and chiefest matter of all is red copper brasse and tinne but very little of the two last such as Canons are made of without any other brickle or harsh metâ—â—â— mingled with it and beeing made of this stuffe let the measures following be observed If the diameter of the calliber or bore be one foot let the morter be two foot of length let the sack or hole for the powder be the third part of a foot broad and halfe a foot deep and let the mortar beare in thicknes an inch and a halfe about one foote high and the rest onely one inch thicke the foot shall bee one inch and a halfe thicke and made square whether the instrument be for the seruice of warres or recreation as doth plainely demonstrate the Figure A following The second mettle is yron such as commonly Ordnance for ships are made of and being cast of this stuffe the rules prescribed shall be found fit onely let the mettle be somewhat â—hicker in euery place The third â—nd fourth and most common materialls are past-boards and canâ—as or pastboords and chord and either or these materialls must bee ioyned together with glue and being to bee made of either of these materialls there must be had a woden foot with the powder sacke or hole made hollow below in the wood as is represented by the Figure B. the precedent measures being obserued which is the third part of the diametre c. But if the instrument be to bee made of either of the two first mettalls that belongs to the Founders office but of the two last materials euery man at his owne pleasure may compose it of what size he please and to begin he must haue a wodden rowler of such bignesse as he shall desire to make the diametre of his Morter-peece and vpon that rowler let the pastbords and canuas with good store of glue be rowled which being done let them dry a while vpon the rowler and another while off from the â—owler and when this kinde of truncke is very dry let the woden foot be ioyned to the one end with glue and nailes very fast and then couer the whole with chord and glue againe which being well dryed the instrument may bee of long service provided that it bee not over-charged and as for the length the longest carry furthest and the thickest dure longest by the figure C. precedent the instrument is represented perfect with the touch-hole in the right place required CHAP. III. The manner how to make granades or mettle for the morter or hand ALl granads being made to breake ought to be composed of the most brickle mettle that may be found as of brasse adding the third part of Tin to it as the Founders know right well they may be also made of yron or of glasse to cast out of mens hands so that the glasse bee made very thicke and these will worke wondrous effects especially in any throng of horsemen or footmen And as for the thicknesse of those which are made of brasse if the diameter be one foot let the mettle be one inch thicke and let them be somewhat longer then they are round as doth demonstrate the figure A. leauing at the one end as it were a handle and on the other a screwed hole by which meanes the granad may be charged also let there be made a hollow vice fitting for the former screw which vice shall be filled with a slow composition made with gunpowder well â—ruised and culled and then made â—nto past with oile of petrole but if the powder be excellent good and strong then mingle with a pound of powder one ounce of Charcoale dust to make it vveaker then let â—he composition bee vvell beaten â—nto the hollovv vice vvhich is represented by the figure B. And the granad shal be filled with fine gun-powder which being full let the vice be screwed into the hollow hole of the granade onely it is to be noted that the vice must not be open at the lower end but haue a bottom sodered strongly wherein shall bee made three or â—â—ure little holes about the bignes of a tag of a point to giue fire to â—â—e powder when the granad hath fâ—owne the space required For the performance wherof it is needfull that you know the quicknesse oâ— slownesse of the mixture wherwith the hollow screw is to be fillâ—d And for the better experience it shall bee conuenient to make of all of one with a granade made 〈◊〉 wood cloath pastboord or aâ—y other stuffe filled with earth â—ut let it be neere to the weight of â—â—ose which you desire to make vse of afterward and by that meanes you may know how long â—ou ought to make your hollow vices which are called by their â—roper tearmes Port-fires and â—hus much of your granade being performed novv let it bee all covered over vvith either Chord Canuas or Pastboord and dipped in glue or in pitch this covering may be neere halfe an inch thicke to the end that the granade going out of the Morter vvith violence it breake not and vvorke his execution vpon your selfe or your company instead of your enemies of vvhat mettall soever the granad is made these rules are to be observed and by the figure C is represented a granade quite finished and dipped into pitch by vvhich meanes it may bee conserved many yeares these sorts of granads vvorke great effects falling vpon houses they cast downe the vvalles and coverings likewise they vvorke vvonderous operations amongst either horsemen or footmen tearing both man and beast in pieces sparing nothing CHAP. IV. The manner how to make Granads of Canuas for the Morter THe operation or execution of these sorts of granads made of Canuas is quite contrary to those prescribed these are onely fit to set a
Towne on fire the houses whereof are most couered with reede straw or broome they are not of so violent execution as the precedent yet notwithstanding of as great cost and for the making of them you must haue a wooden â—owler which is represented by the â—igure A of the bignesse you desire â—o make the granad which alwaies ought to be lesse then the Calliber or bore of your morter to the end â—hey may bee covered afterward â—nd vpon the rowler make a sacke of â—uch cloth as you please as doth reâ—resent the figure B cutting a piece to â—ouer the vpper end when the sacke â—hall be full of the composition folâ—owing Take foure pound of saltâ—eter two pound of gunpowder-â—ust two pound of brimstone all â—hese being well pulverised let them â—e moystened with oyle of petrole â—nd then fill your sackes with the â—ame and cover them with chord which done pierce the sacke full of holes with a great bodkin as representeth the figure E and in euery hole place a little yron barrell charged like a pistoll barrell these little barrells are represented by D which must be driuen into the sacke vp to the head and the granad being thus disposed let there be made at the one end a hole about one inch deepe which shall serue to prime it with powder-dust moystened with oyle of petrole onely it is to be noted that the touch-holes of the little barrells be made somewhat large to the end â—hat the rust stop them not being â—ong time kept amongst the salt-peâ—er and so they may be conserved many yeeres and ready for service â—n all occasions the figure C. doth â—emonstrate the granad perfect and â—urning CHAP. V. How to make fiery arrowes TO avoyde all confusion I will treat first of all the fires which are cast with violence and afterward of those which may be cast out of mens hands And first it is to be observed that fiery arrowes are of great effects and noysome at sea specially if the ships approach any thing neere so that they may bee 〈◊〉 or cast into the sayles chording 〈◊〉 of the ships either by cros-bowes long bowes or any other meanes for so much as a ship being once a fire hardly can it be extinguished And thus the arrowes ought to be made first a long shaft of wood â—nd ioyned to it an yron head made after the manner of the figure A and about the middle of that head make fast a linnen bag in the forme of an oliue leauing open a hole at â—he end before as may be seene by the figure B that it may bee filled with the composition following â—ake one pound of Saltpeter halfe a â—ound of gunpowder-dust and â—alfe a pound of brimstone in â—ovvder all these ingredients â—eeing well mingled and minâ—ed with oyle of petrole then fill â—e bagge round about the arrow â—ead noted by B and then let all be well bound about with wyre And â—r the priming of these arrowes dip cotten weeke into gunpowder wet with water but let the cotton be wel dried again before it be applied to the arrow head Now for the ioyning of your arrow head to the wodden shaft it ought to be so slightly fastned to it that being shooke into any sayle cordage or wood so that if any one would plucke it backe the shaft should slip out of the head and so continue burning in the place pretended and to hinder that one may not pluck out the head with their hands there may be made â— small hole quite through backeward and so the fire will hinder a man from touching any part thereof alâ—hough it should sticke in other of his fellowes clothes There may â—e made little arrows of the same faâ—hion to be cast out of mens hands in â—ny meetings assaults breaches or â—ther occasions and if there shall be â—reat hast for the making of them â—esse fashion may serue hauing onely â—â—ttle stickes of wood about a foote â—nd a halfe long and in stead of the â—ead prescribed a pyke like a â—reat nayle and in lieu of feathers pieces of past-boord stucke into the ends of the wood being slit but the former composition shall be required CHAP. VI. How the granads are to be charged into the morter ALthough it seeme to be but a small difficulty to charge the granads into the morter yet it is the greatest mastery and most curious worke which hath beene found amongst fireworks and the most industrious Enginers that ever I yet saw haue after beene deceived therein and shor â—ut their granads they not taking â—ire which is absolutely one of the greatest faults that may be commitâ—ed Then to avoyde such great erâ—ors it shall be needfull to note maâ—y obseruations following whereof the first is that you put not too much powder in your morter marmarked A but onely the powder-â—ckefull secondly that the priming â—f your granad be firme not spoyâ—ed with dust or dyrtie moistnes or other like thing that it bee not also too dry least all breake about your selfe going out of the Morter taking â—re too quicke which all fire-works â—re very subiect to doe then take â—eed that the granad enter not too â—ardly into the morter which will cause it to breake in the discharging also it is not needful that it enter too loosely All these things being well obserued you must haue alwaies ready port-fires for your morter which may be made about the bignesse of your litle finger and hollowed within about the bignes of a quill even to the bottome noted B this portfire is to enter with a vice into the touch-hole of the morter about half â—n inch and to turne the vice without any key there may be made 4 buttons at the out-side of the vpper â—nd and let this portfire be charged with a slow composition mingling 4. ounces of charcoale with a pound of powder-dust this being done and your granad placed in your morter â—o that it shake not close vp the â—hinks round about your granade within the morter with grease waxe â—itch grease or any such like thing you may so cover the granad with â—ome such stuffe that in any weather wet or dry you shal find no difficulâ—y to shoote off the granad in any place wheresoever you please CHAP. VII The manner how to shoot the granads THere is no lesse difficultie well to discharge the granads then to charge them but the contrary will easily appeare for it is there where the hazards and dangers most great doe meet together And the first of all are the aduersaries Canons from whose dangers the Enginier shall easily conserue himselfe as followeth causing to be made a trench like vnto an halfe moone the connexity thereof being towards the enemy as appeareth by the figure A this trench shall be of such bignesse â—s the Enginier may conueniently place all such things as shall be needâ—ull for his present vse The second difficulty following is how to shoot â—hem right to the places of any Fort â—or
and layd a drying shall be ready for all occasions CHAP. VI. The manner how to make Serpents THe Serpents are to bee made eyther of the composition for rockets on the ground or of that for the ayre for being filled with the composition fitting for the grouund they will spreade and sparkle liuely in the ayre but if filled with the other composition they will fall weaving neerer together notwithstanding eyther sort will shew divers pleasing actions in the ayre being made as followeth Let the cartoush be about foure inches long and rowled vpon a rowler somewhat bigger then a goose quill as is represented by the precedent figure G in the third chapter the paper ought to goe about the rowler nine or ten times and then choaked almost in the middle yet leaving a little hole to see through and the longest part shall be filled with the composition but the shorter with fine grayned powder and choaked close also the longest end must bee halfe choaked close as doth appeare by the figure F chapter third but if you desire not to haue them wamble in the ayre then let them not bee choaked after the composition but as doth represent the figure G both which figures F G represent Serpents quite finished CHAP. VII How to make golden rayne MAny there are specially in France who make rockets yea and boast that they are perfect therein who know not what golden raine is but thinke it to be some other thing then it is wherefore to put them out of doubt and to teach all others who desire the knowledge thereof I will here set downe the description and maner how to make it Take goose quils and cut off the hollow ends leaving them as long as may be as the figure K doth demonstrate third chap. and fill these quils with the composition of rockets for the ayre at the last stopping euery one with a little wet powder to keepe in the dry powder crowning a rocket with these as shall be taught following chap. 12. in its true place will shew a most glorious pleasing raine which some hauing in times past seen haue called it golden rayne for the beauty thereof but of later times it is more commonly called golden hear many beautifull and strange figures may be represented in the ayre with this maner of rayne as shall follow in the 13 chap. treating how to represent many sorts of figures in the ayre with rockets CHAP. VIII The manner how to make Starres ALthough that there bee many sorts of compositions for stars yet I will set downe here but two of the best all the rest being nothing worth but friuolous and expensiue the fiâ—st and best sort is to be made of dry powder and the other of moystened powder as followeth For the â—â—sort take 1. pound of saltpeter halfe a pound of brimston and a quarter of a pound of gunpowder dust al these being pluâ—riâ—ed mingled together wrap the quantity of a nutmeg in tow in a linnen rag or in paper and bind it fast as appeareth by the precedent figure H chap 3 and to prime them you must pierce them with the bodkin and put stoupell or cotten wieke dipt in powder through them which shall be made as followeth in the 11. chap. and to make the second sort take 1. pound of salt-peter and halfe a pound of powder-dust and halfe a pound of brimstone all these wel pulverised and mingled together moysten them with eyther oyle of petrole or els wiâ—h fair water onely to make a past of them wherof make little balls about the bignes of a musket bullet and whilest they are moist rowle them in dry powder dust then let them dry and then may you employ them at your pleasure without further trouble for the last powder in which they are rouled doth serue for their priming This last sort of starres doth not make so beautifull a shew in the ayre as the others for falling downe the flame of them takes the forme of a lampe hauing no force to expel it like wings as the others doe for the flame of the others blowing out of the two sides pierced make it stretch in length and by that meanes shew greater in the ayre CHAP. IX How to make Starres giuing great Reports TO make Starres that each one shall giue a report like a Pistoll or bigger gunne you must first make little saucissons as I taught in the chapter following but the saucisson need not to bee couered with chord and being made and pierced take as much of the former dry composition and bind it to the end of the saucisson which is pierced making a hole through the composition and passe a piece of stoupell or cotten-wieke as in the other starres but if you take of the moyst composition you may onely leaue the paper hollow at the end of the saucisson fitting to contain the quantity of composition required putting a little grained powder before and prime these starres as the others of the same composition these starres are very troublesome and little in vse because that a great rocket can carry but few vp into the ayre and by consequence worke but a small effect and moreouer they are very long in making One may make Starres in the same manner which ending turne to serpents and others as shall please the workeman CHAP. X. The manner how to make Saucissons IN this chapter my intention is not to treate of the saucisson which flyes into the ayre but only of that which stands firme in great workes or else which is applyed to rockets which thus is made as followeth you must haue a rowler of such bignes as you desire to haue the concauity of your saucisson wherupon rowle as much paper as you please and then choake it at the one end which done fill it with grayned powder choake the â—ther end also and cover all the sauâ—isson from the one end to the other with small chord as doth represent â—he figure I. chap. 3. and glue that â—ord with strong glue all ouer and when you would make vse of these â—aucissons pierce them at the end with your bodkin and put into the â—ole a quill filled with fine powder dust which shall serue for a portfire and the other end of the quill shall passe through a board whereupon you meane to fasten them and shall enter into a portfire in the other side of the wood which shall be fastned all along the wood and so may you fasten what store you please neere together or farre asunder this quill is represented by the figure L and by this meanes one end of the 〈◊〉 fire beginning all the whole ranâ— of saucissons will giue their reporâ— one after the other But if your saâ—â—cisson is to be applied to a rocket shall onely be pierced at one end aâ—â— primed with a little grained powdeâ— and fasten it to the top of the rockâ— either with paper parchment or anâ— thing else so that the rocket endiâ—
take the depth of a valley TO take the depth of a Valley there is somewhat more difficult then in the former operations because there are more obseruations to bee made and to begin you must from the point B obserue in some place opposite a â—aâ—ke leuell with the horison as A the leuell whereof you may easilie take at this instrument as is taught in the second proposition by the helpe of the plumbet then from the point B take the distance BC as in the fift proposition or else mecanikelie which being done direct your visuel lines from B to A and to C and leauing the instrument in same state turning your selfe about frame the same angle vpon the plaine which will be FBH then planâ— markes vpon the lines BF and BH and vpoâ— the line BH count as many paces or fathoms as you shall haue found betweene B and C and at the end of your paces 〈…〉 marke which will bee at E thiâ— being done dispose your instrument to make a right angle placing the crosse vpon the â—irst point of the index for the crosse maketh alwaies a right angle when the crosse is vpon the fift point The instrument being thus disposed walke vpon the line BF vntill you may direct your visuell lines by the extreamities of the crosse to B and E or els to FE which will happen in D and no where else then measure the line DE and it will be equall to the depth required G C the demonstration of this proposition is grounded in the 26 of the 1. of Euclide or vpon the equalite of the two triangles BG C and BDE which are both equall and equall angles Although this operation be somewhat more obscure then the others yet I thinke that it is sufficiently explained and therfore I will goe forward to the vse of the sector PROP. IX The manner how to take either distance or altitude with the Sector SVppose the altitude AD were to be taken to the foot whereof you may approch open the Sector 45 adding to it the sights then going forward or backward vpon the line DI vntill you may see the highest part of the altitude A through the two vpper sights the inferior branch of the Sector being paralell to the earth or horison then measure the distance betweene the center of the Sector and the Tower adding to that distance the length of the leg which supporteth your Sector and that shal be the altitude required as appeares in the figure following AB and BC are of equall distance and adding the length of the foot of the sector you will find that it will be height of BD which doth accomplish the altitude of the Tower But to take a distance in any plaine as in the figure precedent the same operation may be vsed except onely the two branches of the sector shall be turned paralell to the horison or ground hauing first made a right angle at the point G or you may operate otherwise first of all prolong a straight line as EGH of what length you please then open your sector to a right angle and set it the point G so that you may see through two sights the point E and where the line visuel of the other two sights strikes along set vp a merke as in I then goe towards it and at your pleasure in the same line set your sector opening it so that you may see the point E and G and keeping your instrument at the same width in the same place place only turne it to the other side so that you may see the point G thorow the sights which before saw E and where the other visuell line shall cut the first line EGH set a marke which will be in H precisely and that shall be the distance required GE the demonstration of this proposition is grounded vpon the 4. and 26. of the 1. of Euclide PROP. X. How to take any distance or altitude inaccessible with the Sector TO obtaine the altitude AB you must first take the distance BC. as is taught by the 9. Proposition precedent and knowing the distance BC which I suppose to be 100 fathom set in the point C your Sector and direct your visuell line through the two vpper sights to the top of the altitude A and let the branch of the Sector be paralel with the ground then leaving your instrument at the same width let fall a perpendicular line vpon the line of the Sector divided into equall parts passing by the 100 number of the inferior branch noted D and note what nuÌ„ber the perpendicular doth cut vpon the vpper branch of the Sector noted H which I suppose here to beâ— the 150 part or number and then set the two points of a paire of compasses vpon the two numbers to wit the one point vpon 100 and the other point vpon 150. of the equall parties then transport the points of the compasse all along one branch of the Sector vpon the line of equall parties and the two points shall denote as many parties vpon the line of the Sector as the Towre doth containe fathoms in altitude adding the length of the foote which supporteth the Sector the demonstration of this Proposition is grounded vpon the 4. Proposition of the 6. of Euclide PROP. XI Of Sines Secants and Tangents BEcause the most noble most artificiall and most certaine way of taking of Altitudes Distances or other Dimentions is by Sines Secants or Tangents I haue set downe their operations in such Propositions as are vsuall in this subiect at the end of this Treatise of Practicall Geometrie and before I enter on the method of operation it is necessary to define what the said Sines Secants Tangents are 1. A righâ— Sine is halfe the Subtence of of the double Arke A Subtence sometimes called a Cord is a right line drawne from any part of the Circumference of a Circle vnto any other part of the same Circumference so the right line DE is the Subtence or Cord of the Ark DGH halfe of which is DF the Sine of the Arke DG and so is MN the Subtence of the Circumference or Arke MGN halfe of it is MI which is the Sine of the Arke MG now the Arke MG is halfe of the Arke MGN and MI is halfe of the Subtence of that double Arke viz. MGN hence it is according to the definition aforesaid that the Sine of any Arke is halfe the Subtence of the double Arke by the same reason OT is the Sine of the Arke BO and MS is the Sine of the Arke BOM and so of others Note further that the totall Sine the Sine of 90 or the Radius is nothing else but the Semidiameter of any Circle viz. CB or CG If a line be drawne touâ—h a Cirâ—lâ— it is called a Tangent a tango as â—he line AB toucheth the Circumference of the Circle in B and so AB is called a Tangent line and is sâ—â—â—uated on the Terme of
head then foure times 8 is 32 and set them vnderneath and the whole will bee 21 32 parts of a fathom which certainely containeth the required superficies CHAP. XI Multiplication of entiers and Fractions TO multiply ¼ by 2½ you must first of al reduce the whole into fractions and then as here aboue multiply numerator by numerator and denominator by denominator and the product will bee 45 â—8 as plainely appeares by the example following But if it were proposed to multiply greater numbers as 20 by 15 26 29 then multiply the 15. intiers by the deno 29. of the fraction then adde the numerator 26 of the same fraction which done will mount to 461 29 then set the 461 ouer a line and the 29. vnder it and afterward multiply the 20 intiers by the 461. which done diuide the product of the whole by the denominator 29. and the numqer required shall bee 317 27 19 as appeares CHAP. XII The diuision of Fractions TO diuide ¾ by â…“ each numerator is to bee multiplied by each denominator opposite and set the product ouer a line aboue them and then diuide the greatest product by the least as followeth CHAP. XIII To diuide intiers and fractions by intiers and fractions TO diuide 12â…” by 3â…™ they must first be reduced all into fractions as before and then you must multiply the numerators by the denominators acrosse as followeth and then diuide the greatest product by the least as this example doth cleerely demonstrate CHAP. XIV Evaluation of fractions which may not be abridged SVppose you were to abridge 7 9 parts of a fathom first you must consider what are the parts of the intier or whole as 6. foot or 72. inches then you must multiply the numerator 7 by the denominator 72 parts and let the product bee diuided by the denominator 9 and then you will finde 56 inches for the eualuation of 7 9 parts of a fathom By this meanes any fraction may be abridged as well in Geometrie as as commerce although they seeme not to be abridged CHAP. XV. For the eualuation of measuring lands YOu must consider that the fathom of 6 foot in length doth containe in superficies 36. and that the 72 inches in length doth contain in superficies 5184 inches and of other measures then to valuate a fraction of 19 4â— parts of a fathom square in superficies you must multiply 5184 by 19. and diuide the product by 47 and there will be 2095 inches for the square of 29 47 of a fathom square and so of other like measures CHAP. XVI Of the rule of three without fractions MVltiply the second number 400 by the third 12. and product 4800 by the first number 4 and the quotient shall be the number required and dispose your rule as followeth months pounds months If in 4 400 12 The probation of this Rule IS to multtplie the first number 4 by the fourth number 1200 and to multiplie the second by the third and the two products will bee equall if the rule bee well made CHAP. XVII Of the rule of three with intiers and fractions FIrst all the intiers must bee reduced into fractions as followeth yards pounds yards If 2¼ 12½ 7½ 9 4 25 2 15 2 Which done you must multiply the second number of fractions as by the third number of fractions 15 then againe multiply the product by 4 the denominator of the first number and then say 2 times 2 or 4 and 4 times 9 is 36 which must bee set vnder the line by which you shal diuide the first product 1500 and the quotient shall be the number required as appeares Heere followeth two examples differing the one from the other whereof the manner of multiplying the one is more easier then the other the first is multiplyed as the precedent but the last is multiplied first by all the intirres viz. by 3 by 8 and by three leauing the fraction â— by it selfe and after all take the thirâ— of the intier viz. of 50000 saying thâ— third part of 5 is one rest 2 for the 10 which is valuated at 20 then say the third part of 20 is 6 and so rests 2 for the second 0 and so to the end and what shall rest at last shall be set ouer a line and your 3 4th or 5 vnder the line then all being added together you shal diuide the product cutting off the figures to the quantitie of the first number saying by ten by a hundred by a thousand by ten thousand by a hundred thousand the remainder is the number required as appeareth 191â…” The first example The second number being mulâ—â—plied by the third doth mount to 57500000 and being diuided by the first multiplyed by 3 as before is taught the quotient will be 191 2 â—0 The second example CHAP. XVIII Extraction of the square roote FIrst dispose your numbers as followeth out of which you meane to draw the roote separating your figures by two and two beginning at the latter end but first strike the halfe circle 73 21 01 and then say the root of 73 is 8 and set 8 before the half circle rests 9 then double the quotient 8 and say 2 times 8 are 16 and set the 6 vnder the last figure of the second part of figures and 1 vnder the first figure of the first part aâ— in this first example Then say how many times is 1 in 9 and it shall be 5. which you shall also set vnder the 1 of the second separation as apppeares in this second example And then say 5 times 1 are 5 which taken out of 9 rests 4 and 5 times 6 are 30 and 30 out of 32 rests 2 and 3 out of 4 rest 1 and then againe say 5 times 5 are 25 which out of 3â— rest 6. and 3. out of 12. rest 9. and then double the quotient and say twice 5 are 10 set 0 vnder 0 of the last separation and keepe â— in memorie and say twice 8 are 16 and 1 that I keepe in mind makes 17 then set downe 7 vnder the 5 and the 1 vnder the 6 of the middle separation as appeares in this example following And then say how many times is 1 in 9 and it shall be 5 times which shall be set downe for quoâ—ient and also vnder the last figure 1 and then say 5 times 1 are 5 out of 9 reâ—ts 4 and 5. times 7 are 35 which out of 36 rests 1 and 3 out of 4 rests 1 and 5 times 0 is 0 and 5. times 5 are 25. out of 31 rests 6 3 out of 10 rests 7 and 1 out of 1 rests 0 and so the rule is ended as appeares following CHAP. XIX Another example of the square root After you haue separated your figures by two and two and drawne 251 the square roote there doth yet rest 268 which must bee reduced into fractions and to begin set that rest 268 ouer a line at the end of your root and that rest