541 gB Wjlffl MMtttT LltSARV -wnivetsfn or University of California • Berkeley Cod. Artuvd,. 34-3. foil vec.JII. imruiunt. ut vo-fcea L#£- St a i tefr-Vtut '(u(¥LcienPt{is wyse hit fonde Thys craft of gemetry yn Egypte londe Yn Egypte he tawghte hyt ful wyde, Yn dyvers londe on every syde. Mony erys afterwarde y vnderstonde Gher }>at f>e craft com ynto J>ys londe. Thys craft com ynto England, as y ghow say, Yn tyme of good kyng Adelstones day." This notice of the introduction of Euclid's Elements into Eng- land, if correct, invalidates the claim of Adelard of Bath, who has always been considered the first that brought them from abroad into this country, and who flourished full two centuries MENSURATION OF HEIGHTS AND DISTANCES. 57 metros. J?at es mesure on englisch. J?an es geome- tri als erthly mesure. for it es one of pe sevene science techyng to mesure J?e erth in heght. depnes and brede. and length. This tretis es departed in thre. £at es to say. hegh mesure. after the "good kyng Adlestone." Adelard translated the Elements from the Arabic into Latin, and early MSS. of the translation occur in so many libraries, that we may fairly conclude that it was in general circulation among mathematicians for a considerable time after it was written. Tiraboschi was the first who remarked, that the translation attributed to Campanus, was in reality Adelard's, with a commentary by the former, (Libri Hist, des Mat. torn. n. p. 48) ; there are also several MSS. in the titles to which this is expressly stated, (MS. Bib. Bodl. Seld. Arch. B. 29. MS. Bodl. 3623. MSS. Paris, Lat. 7213, 7214, 7215. MS. Harl. Mus. Brit. 5266 : " per Adelar- dum Bathoniensum de Arabicoin Latinum translatus,cumCom- mento Campani Novariensis."). In the Library of Queens' Col- lege, Cambridge, there was formerly a MS. entitled "Geometria Euclidis cum Commentario Adelardi" (Lelandi Collect, tom.iii. p. 19), and Chasles (Apercu Historique, p. 509,) mentions one in the library of the Dominicans of St. Marc at Florence, under the same title ; this would appear to intimate that the com- mentary is also by Adelard, and many MSS. in which the name of Campanus does not appear, contain the commentary, (MS. Oxon. in Coll. S. Trin. 30, iv. MS. Harl. Mus. Brit. 5404. MS. Bib. Pub. Cant. Dd. 12, 61). Doctor John Dee, in the Catalogue of his Collection of MSS. (MS. Harl. 1879. MS. Bib. Trin. Coll. Cant. Collect. Gal. O. 4, 20. MS. Ashm. 1142.) gives the title of one, in which the books of Euclid on Optics and Catoptrics, as well as the Elements, appear under Adelard's name. There have been two independent notices of the plagiarism of Campanus ; one by Charles Butler, (the author of the " In- troduction to the Mathematics," 8vo. 1814,) in some MS. papers in the Editor's possession : the other by the Author of the article Geometry, in the " Penny Cyclopaedia." 58 MENSURATION OF playne mesure. and depe mesure. First foryi shewe we hegh mesure. J?at es to say howe any thynge Jmt has heght may be met howe hegh it es. and J?is may be done in many maneres. first J?erfor schewe howe it may be done by pe quad- rant. When you wille wite pe heghte of any thyng j?at you may negh. biholde J?an pe heght of J?at thyng by bothe holes of pe quadrant and come toward and go froward til pe perpendicle f>at es to say pe threde whereon pe plumbe henges falle vpon pe mydel lyne of pe quadrant. J?at es to say pe 45 degre. J?an take als mykel lande behynde ye as fro fethen to pe erthe and marke wele f>at place. J?an mete howe many fete are bytwene fi mark and pe fondement of J?at thyng whos heght you sekest. and sekirly so many fote heght it es. Also when you wilt wite pe heght of any thynge by pe quadrant, biholde pe heght of J?at thyng by bothe pe holes, and byholde vpon what place of pe quadrant pe perpendicle falles. for ouJ?er it wille falle on pe vmbre toward or on pe vmbre froward. and if it falle vpon pe vmbre toward biholde vpon whilk poynte of j?at vmbre pe perpendicle falles. J?an mesure pe distaunce J?at es to say pe space betwene ye and J?at thyng whos heght you sekes, and when you has so done J?an multiply you by 12 j?at same mesure. J?an al J?at comes of J?at multiplyeng departe you by pe nonmbre of pe poyntes of pe vmbre. and to alle J?at comes J?ereof set pe quantite of ]?in heghte. and set al j?is togydere. and J?an you has pe HEIGHTS AND DISTANCES, 59 heght of f e thyng whos heght you sekes. If peraventure f e perpendicle falle vpon f e vmbre froward biholde fan f e poyntes f ereof and torne fern into pe poyntes of pe vmbre toward and do furth fan as we taght byfore and f us sekirly you sale have pe heght of pe thyng whos heght you sekes. namely so pe space be playne bitweene pe and it. Note you fat pe quadrat. fat es to say 4 square whilk es descryvede fat es to say schewed in pe quadrant has tuo sides. fat es to say pe side of pe vmbre toward, and pe side of pe vmbre froward. and af er of f ese 2 sides es depar- ted in 12 even parties. When you holdes pe cone of pe quadrant. fat es to say pe cornel of pe quadrant even vpryght in whilk cornel es pe nayle whereby pe perpendicle henges. fan pe circum- ferens. fat es to say pe cumpasse es toward pe erth. fan fat side of pe quadrat whilk es nere ye es called pe vmbre toward and fat of er side es called f e vmbre froward. and fe 12 departynges of aifer of f o sides are called poyntes. fan es a poynte f e twelft parte of any thyng. namely of ouf er side of f e quadrat in f e quadrant. Also when f e heght of f e sonne es more fan 45 degres. fan f e perpendicle falles vpon f e vmbre toward. And ageynward when f e heght of f e sonne es less fan 45 degrees, fan fe perpendicle falles vpon f e vmbre froward. When sothly f e heght of f e sonne es even 45 degrees, fan fe perpendicle falles even vpon fe 45 whilk es fe medil lyne. If you may noght negh f e thyng fat you wolde 60 . MENSURATION OF mesure for letting of water or summe of er thyng bitwene. fan biholde f e heght f ereof by bothe pe holes, and biholde pe nonmbre of pe poyntes of pe vmbre toward, namely vpon whome pe per- pendicle falles. fan set D for a mark in fat place where you stondes fan. go ferre or nerre fat thyng whos heght you sekes. and fat by an evene lyne and beholde este pe forsaide heght by bothe holes, fan fere stondyng seke pe nonmbre of pe poyntes of pe vmbre toward, and set fere C for anof er mark, fan mesure howe many fete are bitwene p ise tuo markes D and C and kepe fat wele in f i mynde. fan abate pe lesse nonmbre of f ese tuo in f e vmbre toward fro f e more and kepe wele fe difference bytwene fo tuo nonmbres. fan multiplie by twelve fe distaunce bitwene f e forsaide D and C and alle fat comes of fat multeplyeng departe you by f e distaunce of fe poyntes. and to fat fat leves over, set als mykel as fro f ethen to f e erthe and fan sekirli you has f e heght of fat thynge. bot loke f e holes of f i quadrant be right straite and elles you may lightly be deceyvede. Peraventure you standes in an aley. and fe thyng fat you wolde mete es vp on an hegh hille. first fan biholde fe heghte of fe hille by both fe holes of fe quadrant, and fat by tuo stondynges of D and C as we taght nowe next bifore. and marke fat wele in fi mynde. fan biholde in fe same wise fe heght of the hille and of fat same thyng togedire. fan abate fe heght of fe hille HEIGHTS AND DISTANCES. 61 fro al fat remenant and fan sekirly hast ou fe heght of fat thyng. If peraventure fe contry es hilly. fan do fat fe perpendicle falle even vpon pe begynnyng of pe side of pe vmbre froward. fan se by bothe holes in pe thyng to be mesured pe poynte fat es called A and do as we taght bifore. and fat leves after fi wirkyng es pe heght fro A poynte to pe heght of pe thyng : bot fan salt you noght set f ereto pe heght of fi stature. Parcas you woldest mesure pe heght of a thyng by the schadowe fereof. fan abyde til pe sonne be in pe heght of fyve and fourti degrees. fan mesure pe vmbre of fat thyng and fat es pe heght fereof. If you wilt mesure pe heght of any thyng by pe schadowe fat es to say pe vmbre fereof in ilk houre fan do fus. mesure pe vmbre fat es to say pe schadowe of fat thyng. and multiply fat by 12 and al fat departe fan by pe poyntes of pe vmbre toward, and pe nonmbre howe ofte euer it be es pe heght of fat thyng if pe per- pendicle falle vpon pe vmbre toward. If pe perpendicle falle vp pe poyntes of pe vmbre froward. fan multeply pe vmbre. fat es to say pe schaddowe by pe poyntes of fat vmbre. and al fat departe by 12 and fe nonmbre howe oft it be. es fe heght of fat thyng. Or if you wilt, lede f e poyntes of f e vmbre froward into fe poyntes of fe vmbre toward, and fan multeply fe vmbre. fat es to say fe schaddowe of fat thyng by 12 and fan al fat departe by fe 62 MENSURATION OF poyntes of pe vmbre ladde. Whilk after pe ledying are poyntes of pe vmbre toward. Take pe poyntes of the vmbre jms. late pe sonne benies passe by bothe holes, and mark where pe perpendicle falles. J?an counte pe poyntes fro pe begynnyng of pe side of pe vmbre to pe touche of pe perpendicle. and J?o are pe poyntes of pe vmbre. J?at es to say pe scha- dowe. What poyntes ever yei be. whej^er of pe vmbre toward or froward. When any thynge es whose hight you wilt mesure by pe schad- dowe f>ereof and a gerde. j?an rere even vp a gerde vpon pe playne grounde were pe ende of pe schadowe of J?at thyng whilk you wille mesure, so rere it J?at pe one parte of J?ame gerde falle vpon Jmt schadowe. and pe o]?er part of j?at gerde falle withouten. and mark pe place in pe gerde where pe schadowe begynnes to touche it. and by pe quantite of the gerde whilk es bitwene pe touche of pe schadowe in pe gerde and pe playne. multeply pe quan- tite of al pe schadowe whilk es bitwene pe lower party of pe thyng to be mesured and pe toppe of pe schadowe in pe playne. and depart J?an al J?at by pe quantite of pe schadowe whilk es bitwene pe toppe of pe schadowe and pe gerde. and pe nonmbre j?at J?an comes es pe heght of J?at thyng. Also if you wilt mesure pe heght of a thyng by pe schadowe in ilk houre of pe day. take a gerde of two fote longe or thre, and on a playne rayse it even vp. and HEIGHTS AND DISTANCES. 63 fan mesure fe schadowe of it. fan mesure fe schadowe of fat thyng to be mesuride. and multiply fat by pe length of p e gerde. and fan departe al fat by pe schadowe of pe jerde and fat nonmbre howe ofte it be es pe heght of fat thyng. If you wilt have pe heght of any thyng wantyng grounde. as if you be in an house and wolde wite howe ferre were any thyng beyng in pe rofe. Take a table and rayse it vp a litel fro pe erthe. so fat you may se fat ilk thyng bitwene pe erthe and fat table, fan take a reulure and continu it to pe table. fat es to say side to side, and fan se by al pe reulure pe thyng to be mesurede. and fan drawe a lyne in pe table by pe reu- lure. este do pe same in anofer site, fat es to say place of pe table, and make anofer lyne. fan mesure pe heght whilk es bitwene pe hy- here parte of pe table and pe erthe. and fat mesure sale be callede pe heght kept, fan set fe table on fe erthe and take tuo thredes and put fat one in fat one lyne. and fat of er threde in fat of er lyne. fan make a mark fere as f ise tuo thredes metes, fan mesure f e length bitwene fe mark and fe table, and set fereto f e heght kept, and fan hast ou pe heght of fat thyng. If you wilt mesure fe heght of any thyng withouten quadrant and withouten schadowe. rayse evene vp a rodde on a playne ageyne fat thyng and go toward it and froward it til fi sight beme passe by fe heght of fe gerde 64 MENSURATION OP and of pat thyng. J?an loke* howe mikil es bi- twene pi fote and pe grounde of J?at pyng in pe tyme of beholdynge. and to pat length set pe space fro j?ethen to pe erthe. J?an multiply al J?at by pe length of pe jerde. and j?an departe al }?at by pe space bitwene ye and pe gerde and here- with pe quantite fro }?ethen to pe erthe. and J?at es pe heght. Also when you wilt mesure pe heght of any thing by two gerds, even cor- neldly joyned, take a gerde even to pe length of j?i stature, and ano]?er gerde, tuo so longe als J?at in pe myddel of pe lengere gerde, set pe schortere even corneldely fan )?is instrument ]?us made layde by pe playne ground til by pe toppes of bothe pe gerdes you se pe toppe of the J?ing to be mesured, )?an make J?ere a mark and set J>ereto Jri stature, and j?an set pe marke j?ere. and so heigh es J>at thyng howe mekil length es bitwene pe grounde of J?at thyng and pe latter mark, bot forgete noght J?at perpendicle or equi- pendy, J?at es to say, even hangere lolle by pe toppe of pe longer rodde to schewe when J?in instrument es even vpright, and when it bagges Ensample, pe stature of pe matere be called AB, pe gerde doubling it CD pe gerde evene cor- neldy joyned to it AE, and the foundement of pe pyng ¥, J?an I say pe height of the pyng es BF, with pe quantite of BC. When you standes by a walle of a castelle or toure, and you wolde mesure pe heght of it with outen defaute, make 1 MS. loloke. HEIGHTS AND DISTANCES. 65 a quadrat or quarterd. fat es to say a table even foure square of wode or brasse of what quantite you wilt, and ay f e more it be. fe better it es. and loke it be over alle square, in f e manner of ABCD and put a chippe of what length you wilt in fe cornelle B and anofer in fe cornelle C and f e fird in pe cornel D and loke f ei be fast on fat quadrat fat evenly f ei stande raysed vp. and pe side of pe quadrat bitwene A and B mote be persede reulefully. in whilk persyng put a chippe like pe ofer thre. bot it sale be moveable fro A to B and f is chippe sale hight E and wite you that A es pe right cornel vpward. B pe left cornel vpward. C pe left cornel don ward. D pe right cornel donward. When pe face of pe quad- rat es torned toward ye. and f ese chippes I calle eighen as in pe quadrant, fan loke even vp by C and B chippis or eighne to pe hey est of pe toure. So fat pe quadrat joyne to pe walle. and fat highest of pe toure sale hight F loke pe side AB be departed on 30 or 40 or howe fele you wilt, and in pe same manere departe pe side AD. fan move pe chippe E hedire and f edire til you se pe hiest F ageyne thurgh chippes D, E noght chaunging pe raf er place BC fan biholde where pe chippe E stondes bitwene A and B and loke howe mekil fat part es EA to AD and so mekil part es DC to CBF and howe ofte EA es in AB so ofte es CD in CBF forfi multiply DA in AB and fat comes fereof departe by AE and fat nonmbre howe ofte it be es f e heght of CBF and F 66 MENSURATION OF )?at you have al the heght fro F to pe erthe. me- sure pe length AC to pe erth. whilk length set to pe heght CBF and J?at comes is pe verey heght. If you wilt mesure pe heght of any thyng by a myrure. lay pe myrure in pe playne grounde, and go toward and froward til you se pe toppe of J?at thing in pe mydel of J?at myrure. J?an mul- tiply pe playne bitwene pe foundement of J?at thyng to be mesured and pe myrure by pe space fro J?ethen to pe erthe. and J?at comes |?ereof de- parte by pe space bytwene J?i fote and pe myroure and pe nonmbre howe ofte it be es pe heght of J?at thyng. Also als fro J?ethen to pe erthe has it to pe space bytwene jri fote and pe myrure so pe heght of J?at thyng has it to pe playne. Whilk es bitwene pe rote of J?at thyng to be mesured and pe myrure and so ageyne. Nowe we have taght to mesure the heght of a thynge whilk es the first parte of oure tretis. We wil teche to mesure the playne. for that es the se- conde parte. When you wilt mesure pe length of any playne with pe quadrant stonde in one ende of pe playne and byholde J?at opex by bothe holes, and holde pe cone. J?at es to say pe cornelle of pe quadrant nere Jnn eigh. and pe compas toward pe playne to be mesured. ]mn when you sees pe opex ende of J?at playne take pe nonmbre of pe poyntes of pe vmbre froward whilk pe perpendicle kyttes. J?an multeply HEIGHTS AND DISTANCES. 67 pe distaunce. J?at es to say pe space fro J?ethen to pe erthe by 12 and departe ]?at comes J?ereof by pe nonmbre of pe vmbre of pe poyntes froward ra]?er had. and pe nonmbre howe ofte it be es pe quantite of pe length of J?at playne. Parcas pe playne whose length you wolde mesure es noght evene nor even distondyng to Jun orisont. J?at es to say to pe ende of J?in sight, bot J?at playne es lift vp and croked. J?an biholde pe crokidnes hereof by pe holynge. J?at es to say pe eyghne of pe quadrant, whilk j?us you schalt do. set vpright tuo gerdes of one length in pe endes of pe playne to be mesurede. }?an biholde pe toppes of J?ise thynges by bothe pe eyghne of pe quadrant. J?an loke howe fele poyntes pe perpendicle kyttes. and J?at of pe vmbre froward. if parcas pe perpen- dicle falle vpon pe poyntes of pe vmbre toward. J?an torne pern into pe poyntes of pe vmbre fro- ward. J?an kepe phe poyntes. Est se pe ende of }?at playne by bothe pe eyghen of pe quadrant, and loke howe fele poyntes of pe vmbre froward pe perpendicle kyttes. J?an set J?ese poyntes to pe poyntes raj?er kept, if you stonde in a lowere place aftere j?an you did bifore. and if you stonde in an heighere. J?an take away pe poyntes raj?er kept. J>an do with Juse poyntes as you did with pe poyntes in pe next Chapitere bifore in mesuryng of an evene playne. J?at es to say multeply pe dis- taunce fro J?ethen to pe erthe by 12 and depart }?at comes fereof by pe poyntes of pe vmbre fro- ward. and pe nonmbre howe ofte it be es pe f2 68 MENSURATION OF quantite of pe length of J?at playne. When you wilt mesure pe playne of lande or water withouten pe quadrant, take J?an tuo gerdes and rayse }?at one even vp right on pe playne. and calle pe playne BE and pe gerde vp raysed AB in whilk gerde set evene corneldly ano]?er jerde even dis- tonding to pe playne and j?is gerde sale hight CD J?an beside pe gerde vp raised AB set J?in eigh and biholde pe ende of pe playne to be mesurede. and mark by whilk place of j?at o]?er gerde CD }?i sight bem passes, and calle fat poynte E J?an by pe quantite of pe seconde gerde CD multeply pe first gerde AB and departe J?at comes J?ereof by pe quantite of AC and J?an comes pe length of pe playne. When you wilt wite whilk es pe brede of a ryvere. kast a table vpon pe grounde nere pe ryvere. J?an biholde pe ende of J?at oj?er side pe ryver by a reulure vpon pe table. And drawe a lyne by pe reulure on pe table. Est se pe for- saide ende by pe same reulure in an o]?er place of pe table, and pexe make anoj?er line. J?an go in a playne place and lay a threde on J?at one lyne. and anoJ?er threde on J?at oj?er lyne and continu J?em even til J?ei come to gidere. and howe mykel distaunce J?at es to say space es fro pe metyng of pe thredes to pe table so brode es pe ryvere. When you wilt mesure pe brede of a ryvere by a quadrat, make p\ quadrat A BCD as it es saide bifore of mesuryng of heght by pe quadrant. J?an set pe quadrat beside pe ryvere and loke by CB pe ende on J?at oj?qer side pe ryvere. and calle J?at HEIGHTS AND DISTANCES. 69 ende F and B es pe ende on J>is side pe ryvere. J?an go fro B poynte by an evene lyne with J?is quadrat, and J?at pe reulure of J?at lyne be oc- thogonyely. pat es to say even corneldly bytwene B and F til yu se est pe ende F by pe poyntes or chippes D, B in pe quadrat, and calle G pe mark in pe place of pe seconde site. J?at es to say stondynge f>an howe mekil length es bitwene B and G so mekil as pe brede bytwene B and F. Also when you stondes by a ryvere and wille me- sure pe brede of it. put Jris quadrat vpon pe erche nere pe ryvere. and biholde by pe lyne CB pe ende on J?at oJ?er side pe ryvere whilk es called F. fan move E hedire and Jridere til you se pe same ende F by DE noght changyng pe first place of BC. j?an biholde where E stondes bi- twene A and B j?an multiply DA into AB and de- parte j?at comes J>ereof by AE and pe nonmbre J?at comes howe oft it be es pe brede of pe ryvere whilk es BF. When you wilt mesure a playne by a mesure. J?an rayse vp right on pe playne a rod of pe length fro J?in eigh to pe erche. and calle J?at rod AB. on ]?at rod hange a litel myrure. and ay pe lesse pe better, and pe place on pe rod where pe myroure hanges calle C and pe playne to be mesurede sale hight AD. J?an stonde you vpon J?o playne bitwene pe myrure and pe ende of pe length of pe playne. and loke in pe myrure movyng toward and froward pe myrure til you se pe ende of pe playne in pe mydille of the myrure. J?at place of pe playne where you stondes 70 MENSURATION OF you sale calle it E. fan multiply pe length of J?i stondynge fro pe jerde by pe quantite bytwene pe myrure and pe playne. and departe fat comes f ereof by pe distaunce of pe myrure fro pe hyere parte of pe rod. and fat nonmbre howe ofte it be es pe length of fat playne. as f us. multiply AE in AC and depart fat comes f ereof by pe lyne CB. and pe nonmbre fat comes f ereof es pe length of pe playne. Nowe we have taght to mesure the playnes of ilk erthly thyng whilk es the secunde parte of this tretyse. So we teche to mesure the depnes. for that es the thrid parte and laste of this boke. When you wille mesure pe depnes of a welle. loke fro fat one side of pe welle to pe ende of fat of er syde in pe bothome of pe same welle with pe quadrant, and holde pe cornel of pe quadrant nere fine eigh. and pe circumferens toward pe welle and take hede what es pe nonmbre of pe poyntes of pe vmbre toward vpon whilk pe perpendicle falles. fan mesure pe diameter of pe pit or welle. fat es to say pe mouthes brede. and multiply fat by 12 and depart fat comes J?ereof by pe nonmbre of pe forsaide poyntes and howe ofte fat nonmbre be. it es pe depnes of J?at pitte. Also for pe same, rayse vp a table on pe mouthe of pe welle. j?an put vpon pe table a reulure whereby biholde you pe ende on pe bothome J?ereof. fan make f>er a lyne by J?at reulure. est put pe reulure in anof er place HEIGHTS AND DISTANCES. 71 of pe table, and J?an se by )?at reulure pe forsaide ende. and pere make anof>er lyne by pe reulure in J?at table. J>an lay pe table vpon pe playne grounde. and lay on J?ise tuo lynes tuo thredes even til J?ei mete. j?an mesure pe distaunce bitwene pe metyng of ]?o thredes and pe table, and so depe es pe welle Also for pe same.rayse even vp on pe mouthe of pe well swilk a quadrat as we spak of bifore. J?an by CB loke pe ende in pe bothome of pe welle. and J>at sale hight F. J?an move E toward and froward til by DE you se est F noght chaungynge pe first place BC. J?an biholde where E stondes bytwene AB. fan multiply DA in AB and departe J?at comes J?ereof by AE. and J?at nonmbre ho we oft it be es pe depnes of pe welle. whilk es BF. Nowe oure tretis of geometri es thus endid. AN ACCOUNT TABLE FOR THE USE OF MERCHANTS. FROM A MS. OF THE FOURTEENTH CENTURY. Bib. Sloan. Mus. Brit. 213. 0 0 0 0 0 0 0 0 0 0 0 11 10 9 8 7 6 5 * 3 2 1 per yes 0 3 0 0 0 0 0 0 0 0 0 0 0 0 2 0 11 10 9 8 7 6 5 t 3 2 1 0 1 12 schillynges 13 o c 5 a re as — 8" 5- H Ctq re VI — — 0 0 0 0 13 14 15 16 ■ c .a a W X in c X iii c X m c X 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 2 0 17 0 0 0 0 0 0 0 0 0 0 0 3 0 18 0 0 0 0 0 0 0 0 0 0 0 4 0 19 0 0 0 0 0 0 0 0 0 0 0 5 6 — — 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 8 9 This es tabil marchaunte for alle manere acountes. CARMEN DE ALGORISMO. Hjec algorismus ars praesens dicitur ;2 in qua Talibus Indorum3 fruimur bis quinque figuris. 0. 9. 8. 7. 6. 5. 4. 3. 2. 1. Primaque significat unum: duo vero secunda: Tertia significat tria : sic procede sinistra Donee ad extreraam venias, quae cifra vocatur ; Quae nil significat; dat significare sequenti. Quaelibet illarum si primo limite ponas, Simpliciter se significat : si vero secundo, 2 "Haec praesens ars dicitur algorismus ab Algore rege ejus inventore, vel dicitur ab algos quod est ars, et rodos quod est numerus ; quae est ars numerorum vel numerandi, ad quam artem bene sciendum inveniebantur apud Indos bis quinque (id est decern) figurae." Comment. Thomce de Novo-Mercatu. MS. Bib. Reg. Mus. Brit. 12 E. 1. 3 "Hae necessariae figurae sunt Indorum characteros." MS. de numeratione. Bib. Sloan. Mus. Brit. 513, fol. 58. "Cum vidissem Yndos constituisse ix literas in uni verso numero suo propter dis- positionem suam quam posuerunt, volui patefacere de opere quod sit per eas aliquidque esset levius discentibus, si Deus voluerit. Si autem Indi hoc voluerunt et intentio illorum nihil novem Uteris *uit, causa quae mihi potuit. Deus direxit me ad hoc. Si vero alia dicam praeter earn quam ego exposui, hoc fecerunt per hoc quod ego exposui, eadem tam certissime et absque ulla dubitatione poterit inveniri. Levitasque patebit aspicientibus et discentibus." MS. Bibl. Publ. Cant. 1869, Ii. vi.5. 74 CARMEN DE ALGORISMO. Se decies : sursum procedas multiplicands Post praedicta scias breviter quod tres numerorum Distinctae species sunt; nam quidam digiti sunt; Articuli quidam; quidam quoque compositi sunt. Sunt digiti numeri qui semper infra decern sunt; Articuli decupli digitorum ; compositi sunt Illi qui constant ex articulis digitisque. Ergo, proposito numero tibi scribere, primo Respicias quis sit numerus ; quia si digitus sit, Una figura satis sibi ; sed si compositus sit, Primo scribe loco digitum post articulum ; atque Si sit articulus, in primo limite cifram, Articulum vero tu in limite scribe sequenti. Quolibet in numero, si par sit prima figura, Par erit et totum, quicquid sibi continuatur ; Impar si fuerit, totum sibi fiet et impar. Septem4 sunt partes, non plures, istius artis ; Addere, subtrahere, duplareque dimidiare; Sextaque dividere est, sed quinta est multiplicare ; Radicem extrahere pars septima dicitur esse. Subtrahis aut addis a dextris vel mediabis ; A leva dupla, divide, multiplicaque ; 4 En argorisme devon prendre Vii especes .... Adision subtracion Doubloison mediacion Monteploie et division Et de radix enstracion A chez vii especes savoir Doit chascun en memoire avoir Letres qui figures sont dites Et qui excellens sont ecrites. MS. Seld. Arch. B. 26. CARMEN DE ALGORISMO. 75 Extrahe radicem semper sub parte sinistra.5 Addere si numero numerum vis, ordine tali Incipe; scribe duas primo series numerorum Primam sub prima recte ponendo figuram, Et sic de reliquis facias, si sint ibi plures. Inde duas adde primas hac conditione : Si digitus crescat ex additione primorum, Primo scribe loco digitum, quicumque sit ille ; Sed si compositus, in limite scribe sequenti Articulum, primo digitum ; quia sic jubet ordo. Articulus si sit, in primo limite cifram, Articulum vero reliquis inscribe figuris ; Et per se scribas si nulla figura sequatur. Si tibi cifra superveniens occurrerit, illam Dele suppositam ; post illic scribe figuram: Postea procedas reliquas addendo figuras. A numero numerum si sit tibi demere cura, Scribe figurarum series, ut in additione ; Majori numero numerum suppone minorem, Sive pari numero supponatus numerus par. Postea si possis a prima demere primam, Scribas quod remanet, cifram si nil remanebit. Sed si non possis a prima demere primam ; Procedens, unum de limite deme sequenti ; Quod demptum pro denario reputabis ab illo, Subtrahe totalem numerum quern proposuisti. Quo facto, scribe super quicquid remanebit, Facque nonenarios de cifris, cum remeabis, Occurrant si forte cifrae, dum demeris unum ; Postea procedas reliquas demendo figuras. 5 Vide p. 11. 76 CAKMEN DE ALGORISMO. An subtractio sit bene facta probare valebis, Quas subtraxisti primas addendo figuras. Nam, subtractio si bene sit, primas retinebis, Et subtractio facta tibi probat additionem. Si vis duplare numerum, sic incipe ; solam Scribe figurarum seriem, quamcumque voles tu ; Postea procedas primam duplando figuram ; Inde quod existit, scribas, ubi jusserit ordo, Juxta prsecepta quae dantur in additione. Nam si sit digitus, in primo limite scribe ; Articulus si sit, in primo limite cifram, Articulum vero reliquis inscribe figuris ; Vel per se scribas, si nulla figura sequatur : Compositus si sit, in limite scribe sequenti Articulum, primo digitum; quia sic jubet ordo : Et sic de reliquis facias, si sint ibi plures. Incipe sic, si vis aliquem numerum mediare: Scribe figurarum seriem solam, velut ante ; Postea procedas medians, et prima figura Si par aut impar videas ; quia si fuerit par, Dimidiabis earn, scribens quicquid remanebit ; Impar si fuerit, unum demas mediare, Quod non prsesumas, sed quod superest mediabis ; Inde supertactum, fac demptum quod notat unum ; Si monos, dele ; sit ibi cifra post nota supra. Postea procedas hac conditione secunda : Impar6 si fuerit, hinc unum deme priori, Inscribens quinque, nam denos significabit Monos prsedictam : si vero secunda sit una, Ilia deleta, scribatur cifra ; priori 6 u e. figura secundo loco posita. CARMEN DE ALGORISMO. 77 Tradendo quinque pro denario mediato ; Nee cifra scribatur, nisi deinde figura sequatur : Postea procedas reliquas mediando figuras, Ut supra docui, si sint tibi mille figurae. Si mediatio sit bene facta probare valebis, Duplando numerum quern primo dimidiasti. Si tu per numerum numerum vis multiplicare, Scribe duas, quascunque velis, series numerorum ; Ordo turn servetur, ut ultima multiplicandi Ponatur super anteriorem multiplicands ; A leva reliquae sunt scriptae multiplicantis. In digitum cures digitum si ducere, major Per quantum distat a denis respice, debes Namque suo decuplo tocies delere minorem ; Sicque tibi numerus veniens exinde patebit. Postea procedas postremam multiplicando, Recte multiplicans per cunctas inferiores, Conditione tamen tali ; quod multiplicantem Scribas in capite, quicquid processerit inde ; Sed postquam fuerit haec multiplicata, figurae Anteriorentur seriei multiplicantis ; Et sic multiplica, velut istam multiplicasti, Quae sequitur numerum scriptum quibusque figuris. Sed cum multiplicas, primo sic est operandum, Si dabit articulum tibi multiplicatio solum ; Proposita cifra, summam transferre memento. Sin autem digitus excreverit articulusve, Articulus supraposito digito salit ultra ; Si digitus autem, ponas ipsum super ipsam, Subdita multiplicans hanc quae super incidit illi Delebit penitus, et scribens quod venit inde ; 78 CARMEN DE ALGORISMO. Sed cum multiplicat aliam positam super ipsam, Adjunges numerum quern probet ductus earum ; Si supraposita cifra debet multiplicare, Prorsus earn delet, scribi quod loco cifra debet, Sed cifra multiplicans aliam posita super ipsam, Sitque locus supra vacuis super hanc cifra fiet ; Si supra fuerit cifra semper pretereunda est ; Si dubites, an sit bene multiplicatio facta, Divide totalem numerum per multiplicantem, Et reddet numerum emergens inde priorem. Si vis dividere numerum, sic incipe primo ; Scribe duas, quascunque velis, series numerorum ; Majori numero numerum suppone minor em, Nam docet ut major teneat bis terve minorem; Et sub supprima supprimam pone figuram, Sic reliquis reliquas a dextra parte locabis ; Postea de prima primam sub parte sinistra Subtrahe, si possis, quocies potes adminus illud, Scribens quod remanet sub tali conditione ; Ut toties demas demendas a remanente, Quae serie recte ponuntur in inferiori, Unica si, turn sit ibi decet hac operari ; Sed si non possis a prima demere primam, Procedas, et earn numero suppone sequenti ; Hanc uno retrahendo gradu comites retrahantur, Et, quoties poteris, ab eadem deme priorem, Ut toties demas demendas a remanente, Nee plusquam novies aliquem tibi demere debes, Nascitur huicnumerusquociens supraque sequentem Hunc primo scribas, retrahens exinde figuras, Dum fuerit major suppositus inferiori, CARMEN DE ALG0R1SM0. 79 Et rursus fiat divisio more priori ; Et numerum quotiens supra scribas pereunti, Si fiat saltus retrahendo cifra locetur, Et pereat numero quotiens, proponas eidem Cifram in numerum pereat vis, dum locus illic Restat, et expletus divisio non valet ultra : Dum fuerit numerus minor inferiore seorsum Ilium servabis ; hunc multiplicando probabis, Si bene fuisti, divisor multiplicetur Per numerum quotiens ; cum multiplicaveris, adde Totali summae, quod servatum fuit ante, Redditurque tibi numerus quern proposuisti ; Et si nil remanet, hunc multiplicatio reddet. Cum ducis numerum per se, qui pervenit inde Sit tibi quadratus, ductus radix erit ejus, Nee numeros omnes quadratos dicere debes, Est autem omnis numerus radix alicujus. Cum voles numeri radicem quaerere, scribi Debet ; deinde notes si sit locus ultimus impar, Estque figura loco talis scribenda sub illo, Quae, per se ducta, numerum tibi destruit ilium, Vel quantum poteris ex huic delebis eadem ; Et retrahendo duples retrahens duplando sub ilia Quae primo sequitur, duplicatur per duplacatam, Post per se minuens sub ea saliendo. Post his propones digitum, qui, more priori Per precedentes post per se multiplicatus, Destruit in quantum poterit numerum remanentem* Et sic procedas retrahens duplando figuram, Reponendo novam donee totum peragatur, Subdupla propriis servare docetque duplatis ; 80 CARMEN DE ALGORISMO. Si det compositum numerum duplacio, debet Inscribi digitus a dextra parte propinqua, Articulusque loco quo non cedebat duplicando ; Si dabit articulum, sit cifra loco pereunte Articulusque locum tenet unum, duplanda recessit; Si donet digitum, sub prima pone sequente, Si supraposita fuerit duplicata figura Major proponi debet tantum modo cifra, Has retrahens solito propones more figurarum, Usque sub extrema ita fac retrahendo figuras, Si totum debes numerum quern proposuisti, Quadratus fuerit de dupla quern duplicasti, Sicque tibi radix illius certa patebit, Si de duplatis sit juncta supprima figura ; Si radicem per se multiplies habeasque Propositum primo, bene te fuisse probasti ; Non est quadratus, si quis restat, sed habuentur Radix quadrati qui stat major sub eodem ; Vel quicquid remanet tabula servare memento ; Hoc casu radix per se quoque multiplicetur, Et sic quadratus sub primo major habetur, Huic addas remanens, et prius debes habere ; Si locus extremus fuerit par, scribe figuram Sub pereunte loco per quam debes operari, Quae quantum poterit supprimas destruat ambas, Vel penitus legem teneas operando priorem, Si suppositum digitus in fine repertus, Omnino delet illic scribi cifra debet, A leva si qua sit ei sociata figura ; Si cifrse remanent in fine pares decet harum Radices, numero mediam propone partem, CARMEN DE ALGORISMO. 81 Tali quesita radix patebit arte reperta. Per numerum recte si nosti multiplicare Ejus quadratum, numerus qui pervenit inde Dicetur cubicus; primus erit radix ejus; Nee numeros omnes cubicatos dicere debes, Est autem omnis numerus radix alicujus ; Si cures cubici radicem quaerere, primo Inscriptum numerum distinguere per loca debes ; Quae tibi mille notant a mille notante supprima Junctum, summes operandi parte sinistra, Illic et scribas digitum, qui multiplicatus In semet cubicae suprapositum sibi perdat, Et si quid fuerit adjunctum parte sinistra Si non omnino quantum poteris inveniendo, Hunc triplans retrahe saltern, faciendo sub illo Quod manet a digito deleto ; terna figura Sibi propones quae sub triplo asocietur, Et cum subtriplo per earn tripla multiplicatur ; Hinc per earn solam productum multiplicabis, Postea totalem numerum, qui pervenit inde A suprapositis respectu tolle triplatae Addita supprimo cubicae tunc multiplicetur, Respectu cujus, numerus qui progredietur Ex cubito ductu suprapositis adimetur ; Tunc ipsam dele triples saltum faciendo, Semper sub terna, retrahens alias triplicatas Ex hinc triplatis aliam propone figuram, Quae per triplatas ducatur more priori ; Primo sub triplis sibi junctis, postea per se, In numerum ducta, productum de triplicatis : Utque prius dixi numerus qui prevenit inde G 82 CARMEN DE ALGORISMO. A suprapositis has respiciendo trahatur, Huic cubicae junctum supprimo multiplicabis, Respectuque sui, removebis de remanente, Et sic procedas retrahendo triplando figuras. Et proponendo nonam, donee totum peragatur, Subtripla sub propriis servare decet triplicatis ; Si nil in fine remanet, numerus datus ante Est cubicus; cubicam radicemsub tripla probent, Cum digito juncto quern sub primo posuisti, Huic cubicae ducta, numerum reddant tibi primum. Si quid erit remanens non est cubicus, sed habetur Major sub primo qui stat radix cubicati, Servari debet quicquid radice remansit, Extracto numero, decet hoc addi cubicato. Quo facto, nnmerus reddi debet tibi primus. Nam debes per se radicem multiplicare Ex hinc in numerum duces, quod pervenit inde Sub primo cubicus major sic invenietur ; Illi jungatur remanens, et primus habetur, Si per triplatum numerum nequeas operari ; Cifram propones, nil vero per hanc operari Sed retrahens illam cum saltu deinde triplatam, Propones illi digitum sub lege priori, Cumque cifram retrahes saliendo, non triplicabis, Namque nihil cifrae triplacio dicitur esse ; Aut tu cum cifram pertraxeris autem triplicatam, Huic cum subtriplo semper servare memento : Si det compositum, digiti triplacio debet Illius inscribi, digitus saliendo super ipsam ; Quae manet a digito deleto, terna figura ; Articulus jungitur cum triplata pereunte, CARMEN DE ALGORISMO. S3 Sed facit hunc scribi per se triplacio prima, Quae si det digitum per se facit scribi ilium ; Consumpto numero, si sola? fuit tibi cifrae Triplatse, propone cifram saltum faciendo, Cumquecifram retrahe triplatam,scribendo figuram, Propones cifrae, sic procedens operare, Si tres vel duo series sint, pone sub una, A dextris digitum servando prius documentum. g2 PREFATIO DANIELIS DE MERLAI AD LIBRUM DE NATURIS SUPERIORUM ET INFERIORUM. Bib. Arundel. Mus. Brit. 377. Philosophia magistri Danielis de Merlai ad Johannem Norwicensem episcopum. Cum dudum ab Anglia me causa studii ex- cepissem et Parisius aliquandiu moram fecissem, videbam quosdam bestiales in scholiis gravi auc- toritate sedes occupare, habentes coram se scamna duo vel tria et descriptos codices importabiles aureis literis Ulpiani traditiones representantes : necnon et tenentes stilos plumbeos in manibus, cum quibus asteriscos et obelos in libris suis quadam reverentia depingebant : qui dum propter inscientiam suam locum statue tenerent, tamen volebant sola taciturnitate videri sapientes : sed tales cum aliquid dicere conabantur infantissimos ripperiebam. Cum hoc, inquam, in hunc modum se habere Deprehenderem, ne et ego simile dam- num incurrerem, artes que scripturas illuminant DE NATURIS SUPERIORUM ET INFERIORUM. 85 non in transitu salutandas vel sub compendio pretereundas mecum sollicita deliberatione trac- tabam. Sed quoniam doctrina Arabum, que in quadruvio fere tota existit, maxime his diebus apud Toletum celebratur, illuc ut sapientiores mundi philosophos audirem,festinantur properavi. Vocatus vero tandem ab amicis et invitatus ut ab Hyspania redirem, cum pretiosa multitudine librorum in Angliam veni. Cumque nunciatum esset mihi quod in partibus illis discipline libe- rates silentium haberent, et pro Ticio et Seio penitus Aristotiles et Plato oblivioni darentur, vehementer indolui, et tamen ne ego solus inter Romanos Graecus remanerem, ubi hujusmodi stu- dium florere didiceram, iter arripui. Sed in ipso itinere obviam honorem Dominum meum et prin- cipem spiritualem Johannem Norwicensem epis- copum qui me honorifice ut eum decebat reci- piens ; valde meo congratulabatur adventui. PROPOSALS FOR SOME INVENTIONS IN THE MECHANICAL ARTS. MS. Lansd. 101. A Note of sundry sorts of Engynes. 1583. 1. First a carriedge with his properties to carry or drawe fyve hundreth weight with one mans strength. 2. An ingen of wonderfull strength to pull downe parcullices or irone gatts. 3. A chaine of yron non licke it in strength of his bidgnes. 4. A paire of gripes to the same chaing belong- ing of strange fashone. 5. A gine to hoyste or pull vp earth to make rampiors. 6. A scaffolde to be removed. 7. A device to remove any burden of 10 tonne weight without horse or beast. 8. An ingen to lanch shippes. 9. A float to pase men over waters. INVENTIONS IN THE MECHANICAL ARTS. 87 10. A bridg to be carried for passing an army of men ordenaunce and such licke carriges over any ryvers, &c. 11. A myll to grine by water winde or men for forte castell or towne of warr. 12. A crane to hoyst vp 10 or 20 tonne weight. 13. A gynne to hoyst vp any cannon and laie him in his carriage by one man onley. 14. An ingen for clensing or taking away of any shelves or shallow places in the river of Terns or any such river the same device maie serve for clensing of diches about citties or towns pondes or any such licke standing waters. 15. A water myll to rune longer then before tyme. 16. A winde myll and not to turne the howse about. 1 7. To make water workes for fountains cun- ditts and suche licke. 18. To make pipes of lead 6 or 7 foot long without sauder. 19. To make a boat to goe fast one the water without ower or saile.7 7 This was a favourite project about this time. In a few "speciall breife remembrances" of some "pleasante serviceable and rare inventions as I have by longe studie and chargeable practice founde out," addressed to Queen Elizabeth by Ralphe Rabbard, I find one headed " The rarest engyne that ever was invented for sea service;" and this is described as follows: — " A vessell in manner of a Tally or Talliote to passe upon the seas and ryvers without oars or sayle against wynde and tyde, swifter than any that ever hath bynne seene, of wonderfull 88 INVENTIONS IN THE MECHANICAL ARTS. 20. To preserve a boat from drowning and the people that be therein. effect bothe for intelligence and many other admirable exploytes almost beyonde the expectation of man :" MS. Lansd. 121. Now see Bourne's Inventions or Devises, a book deserving the parti- cular attention of all those who are interested in the history of mechanical inventions. Edmund Jentill, in 1594, proposed a " device wonderfull strange," similar to the above : it was one of the inventions which he offered to discover to Lord Burghley, on his release from prison, for counterfeiting foreign coin : — MS. Lansd. 77 and 113. THE PREFACE TO A CALENDAR OR ALMANAC FOR THE YEAR 1430, MS. Harl. 937. My soverayne maistres. certen evydens have done me to vnderstonde jour abylyte to lerne scyens partyculere, and als wele consyder I jour desyre in specyal to lerne a certen conclusyons of J?e new kalender. I say a certen conclusyons for pys cawse for sum of ]?aim profundely to be expressyd or lerned for defawte of termes convenyent in ower moder langage beyn to stronge to a tendyr wytte to comprehende J?at is not elevate be processe and cowrse of scoles. j?erefor als myche as J?e grete phylosophyr sayth he wrappyth hym in hys frende J?at condescendyth to pe ryjtfull prayer of hys frende me and mor mevyth me syth I . . . . bondene to make satisfactyon to jour desire prayng ever dyscrete persone fat j?is redyth or heryth to have my rude endytyng excusyd and 90 THE PREFACE TO my superfluyte of wordes for two cawses. fe furste cawse for so curyos and harde sentens in obscure termes is full tedyos to sy th a tender wy tte to cousayne. pe secunde cawse is for sothely me semythe better to wryte and twyse teche one gode sentens fen ones forgoten. levyng ferfore all vayn preambles of superfluyte fat papyr ful- fyllygte with owtyn fruyte f is lesone I gyfe gow fyrste fat in gour gerus beyn xii moneth. January February March Aprile May June July Auguste Septembre Octobre Novembre Decembre. And ccc and sexty days and 5 and sex odde howres qwyche odde howres gedyrd togeder 4 gerus makyth 24 howres fat is a day naturale so fat in p is gere fat is clepyt lyp gere beyng ccc and sexty and sex days. In everych gere we vse a new letter fat Sonday gothe by. In fe lyp gere we occupy twene f e furste servyth fro new gerus day to seynt mathye day. f e secunde tellyth us owre soneday f e remnande of f e gere. gour pryme schal be to gow a specyal doctour of dyverse conclusyons querfore in latyn he is clepyt f e golden nownbur and begynnygt at one and rennyth to xix and turnyth agayn to one, and so in case fat one were pryme f is gere next gere schulde 2 be pryme and f e 3 gere 3. so rennyng gere be gere to xix and agayn to one and so abowte with owten ende. In f e fyrst table of gowre kalendere by f e reede letters in f e firste lyne joyned to f e blake letters in f e secunde lyne schal ge knowe f e lyp gere qwen it fallyth. In A CALENDAR FOR 1430. 91 f e 3 lyne beyn wryten primes frome one to xix and turnyth agayn. Seche fen jour prime of pe gere fat ge be in jonyd to pe domynycale letter of pe same gere and fen may ge se by pe rede letters in pe fyrst lyne qwenus it be lyp gere, or how nere ge be, fis ensample. I gyffe gow pe gere of owre lorde a.mccccxxx in qwych gere fis kalendere to gow was wryten. Went prime by 6 and gowre letter domynycale was A qwyche beyn jonyd and wryten to geder in pe hede and pe begynnyng of fis sayd table. fen behalde how pe next gere folowyng schal 7 be prime and G domynycale. And pe next gere after schal 8 be prime and be lyp gere as ge may se by pe reede letter F jonyd to pe blak letter E so fat in fis lyp gere schal F be domynycale letter fro pe Circumscycyon fat is clepyd newgerus day vnto seynt Mathye day. and fen schal E be gour sonday letter to pe gerus ynde as I sayd beforne. J?is doctryne kepe Jmrghe alle pe forsayd table. Amonge pe blak letters in pe same lyne beyn wryten reede letters qwyche teches pe indyccyon fat is a terme ful necessary to J?aim fat lyste knaw pe verray and certene date of pe Pope bulles and of pe olde Imperyale wrytynge of Rome, gyf it lyke gow to wytte in qwat Jndyccyon ge be in Take gour prime and gour domynycale letter of pe same gere fat ge beyn in qwyche letter in case fat it be rede calle fat f e first Jndyccyon and fe next letter fe secunde jndiccion and so rennyng vnto xv and gen agayn to one at fe 92 THE PREFACE TO next reede letter, jif it be a blake letter J?at gour prime fallyth one cownte fro pe next goyng beforus and so mony is pe jndiccyon J?at $e seche forgete not pis lesone. A table J?at next folowygt is callyd pe table of pe 5 festes moveyabylle J?at is to say qwen septuagesime commeygt in Jmt is pe sonday be- fore lentyn qwen Allia is closyd vp. Also {?is table tellygt qwen lentyn fallyth qwen Eysterday. qwen pe Rogacyons and qwen qwytesoneday. It techygt also how mony weykes be fro Crystemes- day to lentyn how mony wekes and days fro qwytesonday to mydsommerday And how mony fro Whytesonday to pe advente on jris maner. Tak pe prime of pe jere J?at ge be in. in pe fyrste lyne wryten with reede. and pe domynycale letter next folowyng in pe secunde lyne. and folowe stryght forthe in to pe lyne of J?at thynge wryten in pe heyde Jmt ge desyre. and by pe nownbur )?at ge fynde schalle ge knaw J?at ge desyre. Vnderstonde wele j?is ensample. Aftyr crystynmesse pe gere of owre lorde a.m.cccc.xxx. I was aferde of lentyn and lokyd in my kalender for septuagesime and lentyn how nere J?ai were and soghte my prime in pe fyrste lyne J?at hap- pynd to be 7 pe same tyme. and toke my do- mynycale letter J?at was G next following. J?o turnyd I forth fro J?is G streght in to nexte lyne J?at haythe Septuagesime writyn in pe hede and I fonde 28 J>ere wryten J?ereby demyd I J?at septuagesime sonday schal falle ge 28 day A CALENDAR FOR 1430. 93 of pe moneyth of Januare wrytyn in pe hede. j?o loked I forthermore in pe next lyne that hayth lentyn wry ten in pe hede and J?ere I fonde 18 wryten and J?ereby I demyd f>at first sonneday of lentyn schulde be on pe 18 day of pe moneyth of February nexte wrytyn above £yt I loked for- thermore in pe next lyne for Esterday and fonde pere 1 wryten and J?en wyste I J?at Esterday schulde be on pe firste day of pe moneythe of Aprile wrytyn next above, and jow take ensample as I have do and loke je kepe pe same thurghe alle pe table. JOHANNIS NORFOLK IN AETEM PROGRESSIONS SUMMULA, MS. HarL 3742. In artem progressionis continue et discontinue secundum magistrum Johannem Norefolk incipit Non invenientes sed doctrinam tradita inde numerorum progressione ab Algore rege quondam Castellie suo in Algorismo de integris perficere curantes. occultas aliorum tradiciones si que fu- erint penitus omittentes dummoda tales nostra assidua percunctacione non obstante re maxima nobis miranda ante nostram edicionem in nostras manus visuras nos nusquam recepisse fatemur testimonium tamen fide dignorum nostrorum ami- corum nobis possibilitatem operis concludebat eo quod tales regulas utilem artem progressionis concernentes ante nostra tempora fuisse editas testabantur. Oportunum igitur apud nos indi- cavimus cum ars tarn preciosa arithmeticeque sciencie prefulgida margarita conspectibus ho- minum se non presentabat sed quorundam forte inpericia incarcerata atque porcis oblata sciolos IN ARTEM PROGRESSIONIS SUMMULA. 95 latebat circa possibilitatem operis novarumque regularum edicionem divino auxilio primitus in- vocato petitisque auctoritatibus et favoribus eorum quibus incumbit hujusmodi summulas corrigere operas exiles soleritur impendere et eas sic collectas nostro calamo rudique stilo de- scriptas. Primo ad honorem Dei beatissime virginis Marie et omnium suorum sanctorum ac ad profectum animarum omnium fidelium defunctorum quarum collegio Oxonie primus vicecustos exciteramus et ad ceteras hujusmodi scienciam diligentes plenius promulgare cura- vimus. Reverendam ergo tuam benevolenciam quiscunque es o carissime has nostras regulas exiles exiliterque collectas placito vultu bonoque animo ut acceptare digneris benigne rogamus. Est namque opus harum regularum omnium nu- merorum proportionalitate aliqua comperatorum sive plures paucioresve fuerunt ut in his exem- plis. 1. 2. 4. 8. 16. 32. vel sic. 5. 6. 7. 8. vel sic. 4. 8. 12. 16. vel sic. 1. 2. 3. 4. 5. 6. et sic ad quo- titatem unitatum sive percussionum horologii quas sonat horologium ad 12 horas vel ad quar- tern partem diei naturalis et similibus quid re- sultans fuerit levi arificio explicare. Fiat obse- cramus legitimum kalendare ergo in anima tua o lector honorande hoc scriptum exiguum magna tamen solertia et industria compilatum nee desit tibi spes enarrandi quid ex predictis numeris re- sultat et omnibus consimilibus proportionalitate aliqua comperatis cum regule satis sufficienter 96 JOHANNIS NORFOLK sunt tradite et nomen auctoris publice prefato est monstratum preambulo. Sic ergo karissime incipe et mente tua recollige tria documenta generalia que tractatus presentis processimi ple- nissime facilitabunt. Primum documentum est quod scias quid progressio sit et quod sectionem patitur in continuam et discontinuam prout pos- terius apercius clarescet. Secundum documentum est ut bene consideres quod omnium numerorum ordinate se in progressione habencium aliqua proportionalitas reperta nee quid ex talibus ar- tificiose poterit certificari propter infinite vari- abilitatis talem. Tercium est quod non deficiat tibi spes perveniendi in finem premissum per hanc artem facillimam cum de omnibus numeris proportionalitate geometrica ordinatis ac de qui- busdam arithmetica proportionalitate magis fa- mose reperibilibus dantur regule sequentes solum omittentes numeros inusitatos arithmetica tamen proportionalitate constitutos ac omnes numeros musica proportionalitate collectos turn facilius sit enarrare qui ex talibus numeris resultat quam hujusmodi numeros artificiose reperire et com- ponere et hec tria documenta nota diligentissime. Specificantes modo predicta tria documenta ge- neralia dicamus primo quod progressio est nu- merorum diversorum ordinata collectio et dicitur ordinata pro tanto quod si aliqua collectione numerus minor sequatur majorem ut hie. 12 65. 413. ibi non est ratio progressio judicanda quamvis enim metaphorica progressio nuncupanda IN ARTEM PROGRESSIONIS SUMMULA. 97 €st omnis numerorum collectio a minore inci- piendo terminando in majorem numerum ipsa tamen proprie dicitur progressio cum numeri quadam proportionalitate colliguntur istud nam- que evidentissimum est cum apud omnes maxime indicatur progressionem naturalem esse incipiendo computare ab unitate ascendendo ad alios nume- ros naturali ordine succedentes et ibi aliqua pro- portionalitas inter terminos reperta est et que postea apparebit. Progressio sic proprie sumpta dividitur in continuam et discontinuam progressio continua est numerorum diversorum ordinata collectio nullo numero alium superexcrescente nisi sola unitate et ista potest indrunter contin- gere vel incipiendo ab unitate vel ultra unitatem ut sic. 1. 2. 3. 4. 5. vel sic. 3. 4. 5. 6. et caetera. Progressio vero discontinua est numerorum di- versorum ordinata collectio uno alio supraex- crescente plusquam unitate servata tamen ut prius in progressione continua proportionalitate geometrica et hoc indrunter incipiendo ab uni- tate vel altro numero quocunque ut hie. 1. 4. 7. 10. 13. et caetera. vel sic. 5. 7.9. 11. 13. 15# et caetera. Et quia progressio proprie dicta ut dictum est constat ex numeris geometrica pro- portionalitate covintis ne tractatus iste neces- sariis videatur deficere superfluisve habundare quorum utrumque collectoris impericie ascri- bendum esset videndum est de proportionalitate quid sit et ejus speciebus quae etiam species ejus, ad hanc artem magis pertinet que vero abji- H 98 IN ARTEM PROGRESSIONIS SUMMULA. cienda. Est namque proportionalitas duarum proportionum aut plurimum in simul compera- torum habitudo hujus autem tres sunt species scilicet geometrica arithmetica arismonica sive musica proportionalitas geometrica est quando sunt tres termini aut plures et equales est ex- cessus secundi ad primum sicut tertii ad se- cundum et quarti ad tercium et caetera si tanti fuerint ut hie. 1. 2. 3. 4. 5. 6. vel sic. 2. 4. 6. 8. vel sic. 2. 7. 12. 17. 22. et caetera. Proportio- nalitas vero arithmetica est quando sunt tres ter mini vel plures et equalis est proportio inter primum et secundum et secundum ad tertium et tercium ad quartum ut hie. 1. 2. 4. 8. 16. vel sic. 1. 3. 9. 27. et caetera. Sed proporcionalitas mu- sica sive arsmonica est quando sunt tres termini insimul proporcionalitati et equalis est proporcio tercium ad primum sicut est proporcio excessus tercii ad secundum ad excessum secundi ad primum ut hie. 3. 4. 6. vel sic. 6. 8. 12. vel sic. 12. 16. 24. hiis sic brevissime recitatis per nos dicamus quod in omnibus numeris propor- cionalitate geometrica proporcionalitatis hec ars habet locum sive plures fuerint sic pauciores eciam in numeris famosioribus arithmetica pro- porcionalitate comperatis ut in numeris propor- cionalitate duplica collectis omittentes alias spe- cies ejusdem proporcionalitatis sicut omittimus omnes species arsmonica proporcionalitate inte- gritas ob id quod superius docetur documento tertio. Hiis sic specificatis inducamus primam IN ARTEM PROGRESSIONIS SUMMULA. 99 regulam primo de numero geometrica proporcion- alitate adunatis et ubi progressio continua est que hec est. Numero locorum secundum progres- sionem continuam exeunte sub numero pari per minus medium multiplicetur numerus locorum et habetur quod queritur haec namque regula gene- ralis est ut ostendent exempla posita in prima tabula in fine tractatus hujus et quamvis hsec re- gula viris maturis aliqualiter arithmetice sciencie noticia inbutis satis lucid a est ut tamen juvenes et minus provecti per earn sapiant sic earn delu- cidamus ut ejus aliarumque regularum recitan- darum noticie perfcctus habeantur primo notanda sunt quid nominis horum quinque terminorum scilicet numerus locorum vel loca numerorum numerus par numerus impar majus medium minus medium eiis quinque terris nostro more declaratis facilime perquiritur sensus regule pre- cedentis ac cuj usque quatuor regularum sequen- cium. Sic vero incipientes dicamus quod non probabile aliquem pro eadem morula sive eodem .... plures numeros recitare et sic numerum primo recitatum dicimus esse in primo loco et secundo recitatum numerum dicimus esse in secundo loco et sic de aliis quibuscumque reci- tandis conformiter sicut si numeri diversi scribi debeant oportet quod variis locis scribantur et de eiis idem indicare quoad loca quare citissime cognoscitur numerus locorum secundum numera- cionem numerorum enumeratorum. De secundo et tercio ternimis in simul licet annotare quia 100 IN ARTEM PROGRESSIONS SUMMULA. cognito quid sit numerus par tanquam suura privatum cognoscitur numerus inpar qui aliam cognicionem non habet quam per suum positivum ut docet aristolus. Est autem numerus par ut notat Boycius in sua arithmetica et est sua prima definitio illius quilibet numerus qui potest in duo equalia nullo medio intercedente dividi ab hac namque condicione deficit numerus impar ut constat quia quamvis quinarius dividi potest in duos binarios etiam in duo equalia tamen aliqua unitas integrans quinarium intercidit cum duo et tria faciunt quinque. Ceterum restat declarare quid minus medium quidve majus medium sit promissum ut impleatur pro quorum noticia est primo animadvertendum quid sit medium. Est namque medium quod equaliter distat ab extremis et sicut facilime ymaginatur medium ubi loca numerorum numerantur numero impari ut hie 1. 2. 3. 4. 5. 3. est medium vel sic ubi loca numerorum numerantur a numero pari saltern ultra binarium inveniendi sunt duo tales numeri quorum minor numerus mius medium dicitur et major numerus majus medium nuncupatur ut hie. 2. 3. 4. 5. numerus locorum est quatuor minus medium. 3. majus medium. 4. aut sic. 5. 8. 11. 14. 4. est numerus locorum ut prius minus medium est. 8. majus medium est. 11. hiis sic cognitis evidens est prima regula prius recitata de progressione continua et sequitur secunda regula de progressione continua eadem facilitate que hec est. Numero locorum secundum pro- IN ARTEM PROGRESSIONS SUMMULA. 101 gressionem continuam existente sub numero impari per medium multiplicetur numerus lo- corum et habetur quod queritur sine aliqua sub- traction vel additione et hec regula certissima est perfectus a nobis jam duabus regulis ad progressionem continuam pertinentibus succe- dunt. 3. regule spectantes ad progressionem discontinuam et quia in tali progressione minor erat difficultas nobis certitudinem regularum uti- ]ium invenire ut tamen posteri nostri eas regulas citissime cognoscant quas nostris magnis laboribus a diu percuntavimus licet ipse aliquibus obscuris terminis a nobis primo dentur declarationibus tamen nostris levissime conceptis quilibet me- diocri ingenii capacitate fultus ipsas poterit faci- lime experiri et est prima regula talis. In pro- gressione discontinua numero locorum exeunte sub numero pari et si eorum numerorum excessus nominatus fuerit a numero impari per majus medium intelligibile inter numeros mediates mul- tiplicetur numerus locorum et subtrahatur medie- tas numeri locorum et habetur quod queritur hanc autem regulam assumpto hoc exemplo. 5. 10. 15. 20. sic delucidamus nam in exemplo dato numerorum loca esse sub numero pari nul- lius ambigit et unumquemque numerum exce- dere suum numerum in medietate precedentem per 5. 3. qui est numerus imper nulli dubium est tunc constat quod inter duos numeros media- les qui sunt 10. et 15. quatuor numeros ordine naturali intelligi posse videlicet 11. 12. 13. 14. 102 IN ARTEM PROGRESSIONS SUMMULA. eorumque numerorum subintellectorum majus medium esse 13 hie vero numerus 13. multi- plicandus est per numerum locorum scilicet 4. et subtrahatur 2. qui est medietas numeri lo- corum et patet summa ideo vero dicitur 13. medium intelligibile quia non expresse inter numeros exemplo positos invenitur sed quadam solertia per intellectus indagacionem experitur ac eciam quod majus medium sit consimiliter indicatur. Hiis sic declaratis sequitur secunda regula que hec est. Si progressione discon- tinua numerus locorum fuerit par eorumque numerorum excessu exeunte sub numero pari per medium intelligibilem inter numeros medi- ales multiplicetur numerus locorum et habetur summa quesita hanc autem regulam duobus exemplis declaramus sic primum exemplum. 2. 4. 6. 8. in quo exemplo medium intelligi- bile est 5. qui ductus in numerum locorum exurgunt 20 ut evidet qui est summa totalis. Exemplum secundum prefate regule in quo exemplo numerus excessus est 4. medium vero intelligibilem 11. cum inter duos numeros mediales scilicet 9. et 13. ordine naturali in- tercipiuntur hii tres numeri scilicet 10. 11. 12. aut intelligi possunt quorum medium est 11. ut dictum est per quod multiplicetur numerus locorum et habetur numerus ex omnibus resul- tans. Ultima regula secundum genus propor- cionalitatis geometrice et habita in ordine est hec progressione discontinua numero locorum ex- IN ARTEM PROGRESSIONS SUMMULA. 103 eunte sub numero impari per medium multi- plicetur numerus locorum et habetur quod que- ritur exemplum 6. 8. 10. per 8. multiplicetur numerus locorum qui est tercia et exurgunt. 24. Et quia prius promissum est regulas fieri de quibusdam numeris arithmetica proporcio- nalitate comparatis hanc solam regulam dispo- suimus pro numeris magis famosis et sunt illi numeri qui proporcione dupla inter se comperan- tur ut hie 1. 2. 4. 8. 16. 32. et cetera et est regula talis. Si plures numeri proporcionalitate arith- metica comperati ubi posterior suum inmediate priorem proporcione dupla excedit insimul com- perentur dupletur ultimus et a resultante sub- trahatur primus et habetur quod queritur evidet hec regula in exemplo posito nam dupatur 32. resultant 64. a quibus subtrahatur unitas que est primus numerus et resultant 63. et hec regula generalis est in omnibus talibus sive ab unitate incipiatur sive non sive vero loca numerorum paria fuerint sive non de aliis autem numeris secundum hoc genus proporcionalitatis dicimus nunc ut dictum est prius documento tertio. et sic perfectus est iste tractatus brevissimus in collegio animarum Oxonie anno domini millesimo quadrin- gentesimo quadragesimo quinto quod Norfolk scrip- tor ac compilator hujus tractatus. Explicit. Nil amplius restat ad perfeccionem hujus sum- mule colligere et scribere quam quod anteadictum 104 IN ARTEM PROGRESSIONS SUMMULA. est sed solum regulas prescriptas ad facilitatem prospicere volencium recolligere cum tabulis exemplarium ac consulere quod lectores ejusdem perfecte sint in arte multiplicationis per intel- lectum ideo sequitur tabula subscripta et deo gratias. Prima regula. Numero locorum secundum progressionem continuam exeunte sub numero pari per minus medium multiplicetur numerus locorum et ad- datur medietas numeri locorum et habetur quod queritur. Exemplum prime regule. 1.2.3.4.5.6.7.8.9.10. 11 . 12. 2.3.4.5.6.7. vel sic 6.7.8.9. 10.11. 6.7.8. vel sic 19.20.21.22. VelsiCi3'.4'.5*. Secunda regula. Numero locorum secundum progressionem continuam exeunte sub numero impari per medi- um multiplicetur numerus locorum et habetur quod queritur sine aliqua subtraccione vel addi- cione. Exemplum secunde regule. 1 . 2 . 3 . 4 . 5 . vel sic 5 . 6 . 7 . 8 . 9 . 10 . 1 1. vel sic 12. 13. 14 . 15 . 16. vel sic 21 . 22 . 23 . 24 . 25 . 26 . 27. IN ARTEM PROGRESSIONS SUMMULA. 105 Tercia regula. In progressione discontinua numero locorum exeunte sub numero pari et si eorum numerorum excessus nominatus fuerit a numero impari per majus medium intelligibilem inter numeros medi- ales multiplicetur Humerus locorum et subtra- hatur medietas numeri locorum et habetur quod queritur. Exemplum tercie regule. .5.10. 15. 20. 25.30. vel sic 2.5. 8. 1 J. velsic 3. 10. 17. 24. 31.38. vel sic 3.6.9. 12. Quarta regula. Numerus locorum si progressione discontinua fuerit par eorumque numerorum excessu exeunte sub numero pari per medium intelligibilem inter numeros mediales multiplicetur numerus locorum et habetur quod queritur. Exemplum quarte regule. 5.9. 13. 17. velsic 3.5.7.9. vel sic 2 . 4 . 6 . 8 . 10 . 12. vel sic 3 . 7 . 1 1 . 15. Quinta regula. Numero locorum secundum progressionem discontinuam exeunte sub numero impari per medium multiplicetur numerus locorum et habe- tur quod queritur. Exemplum quinte regule. 4.6.8. vel sic 3. 5. 7.9. 11 . 13. 15. vel sic 12.14.16. velsic 3 . 7. 1 1 . 1 5. 19 . 23 . 27. 106 IN ARTEM PROGRESSIONIS SUMMULA. Sexta regula sed est de numeris arithmetica proportionalitate collectis. Si plures numeri ubi posterior suum immediate precedentem proportione dupla excedat insimul comparentur dupletur ultimus et a resultante subtrahitur primus et habetur quod queritur. Exemplum sexte regule. 12 . 4 . 8 . 16 . 32. vel sic 2.4.8. 16. Explicit tabula exemplarium summule super progressione continua et discontinua secundum Norefolk. APPENDIX. I. A few Observations on the Numerical Contractions found in some manuscripts of the Treatise on Geometry by Boetius. The remarks which follow have not, as far as I am aware, found a place in any prior publication ; I intend them to form an appendix to the interesting chapter on the same subject presented to the literary world by M. Chasles. I could not have connected the following pages in the form of a continuous history, without introducing much that is already known ; I have, therefore, considered it advisable to place my notes under distinct articles, with- out any attempt at arrangement. 1. It is very probable that the well-known passage on the Abacus, in the first book of the Geometry of Boetius, is an interpolation. For in a MS. once belonging to Mr. Ames, no such passage appears ; and in another, now in the library of Trinity College, it is also wanting : again, no such contractions occur in any copy of the Treatise on i2 108 APPENDIX. Arithmetic * by the same author ; although, in the library f just mentioned, there is a list of them, on a fly-leaf to a MS. of that work, in a hand-writing of the fourteenth century, which is thus headed: Primus igin ; andras ; ormis ; quarto subit arbas ; Quinque quinas ; termas ; zenis ; temenias ; celentis. and over these names the contractions are written, as well as Roman numerals explaining them. 2. There are two MSS. in the Bodleian Library which merit particular attention. One, MS. Hatton. 112, pos- sesses two distinct treatises on arithmetic on this system : the first is very extensive, but anonymous; the rubrication to the preface of the other is as follows : Incipit pre- fatio libri Abaci quce junior Berhelinus edidit Parisiis, Domino suo Amulio. In both these treatises, as well as in the other MS., local position is clearly pointed out. 3. Vossius+ attributes them to a Grecian origin; Huet§ derives them from the Hebrew; and the Bodleian MSS. refer them to Syria and Chaldea. It is scarcely neces- sary to observe, that there is no connection between these numerals and those among the contractions of Tyro and Seneca. 4. M. Chasles has confused the sipos and celentis, the latter of wrhich was seldom used as a cipher. In the second Bodleian MS. we read, inscribitur et in ultimo figura 0, sipos nomine. Que licet numerum nullum signified: turn ad alia qucedam utilis est. In the * Most of the MSS. of this work that I have examined are very old, — generally prior to the thirteenth century. Only one MS. that I am ac- quainted with (Bib. Burn. 275.) contains Arabic numerals. It may also be remarked here, that a treatise on arithmetic in verse, by one Leopald (Bib. Arund. 339.) possesses numerals whose forms are, as far as I know, unique. But this tract will receive its due attention in a proper place. t R. xv. 16. There are also a few pages on arithmetic, which contain the following account of its rise among the ancients : Hanc igitur artem nume- randi apud Grecos Samius Pitagoras et Aristoteles scripserunt, diffusiusque Nicomachus et Euclides ; licet et alii in eadem floreunt, ut Erastosthenes et Crisippus. Apud Latinos primus Apuleius, deinde Boecius. % Observationes ad Pomponiam Melam. 4to. Hag. Com. 1658, p. 64. § Demonstrate Evangelica. Prop. iv. p. 173. E. APPENDIX. 109 Lansdown collection (842) in the British Museum, is a very beautiful MS. of the whole works of Boetius : what renders it more interesting in the present inquiry, is the contraction for the sipos without the drawing of an abacus, which curiously illustrates the difficulty of the transition from numerical operations, by means of that in- strument, to local position without distinguishing boun- daries. 5. The Metz MS. in the Arundel collection, referred to by Mr. Hallam, is probably an abridgment of one or more extensive treatises on the subject : the author says, quicquid ab abacistis excerpere potui compendiose collegi, componens inde mihi certas regulas, que volentibus ad hanc disciplinam attingere non inutiles. He quotes Boetius. 6. The following verses occur in a MS.* of the four- teenth century on arithmetic : Unus adest igin ; andras duo ; tres reor armin ; Quatuor est arbas ; et per quinque fore quinas ; Sex calcis ; septem zenis ; octo zenienias ; Novem celentis ; per deno sume priorem. And a list of the contractions is given on the preceding page of the same volume. 7. The fractional notation appears to be as curious as the integral, but the contractions are not quite so arbi- trary, and a regular system is evidently followed up throughout. It is for the most part merely an adapta- * Bib. Trin. Coll. Cant, inter MSS. Gal. O. 2. 45. f. 33b. Chasles has given verses to the same import, but he does not mention the source whence he has obtained them. There is every reason to think that the sipos was a later improvement, and the Metz MS. contains no allusion to it. Mr. Barn- well, of the British Museum, informs me, that a short tract on celentis was once pointed out to him in a MS. volume belonging to that establishment, but no reference to it being in any of the catalogues, I have not been fortunate enough to find it. 110 APPENDIX. tion of the Roman weights to numerical computation ; for instance, taking as for unity, we have— H deunx ±% dextans ^2 dodrans T8T bisse ■£$ septunx T% semis T5T quicunx T\ triens T\ quadrans T\ sextans yz uncia The uncia was also divided into twelve portions, but differently — i 2 semiuncia £ duella 1 sicilius £ sexcula 1 T dragma TV hemissecla tV tremissis -2lT scrupulus A obulus TV bissiliqua A ceraces TJT siliqua To these was added the iT2th Part of tne uncia : ut usque ad minimum extremum diatessaron et diapente sympho- niarum tonorum semitonorumque intervallis distinclarum, harum fractionum denominatio conscenderet vel conten- deret. Zambertus gives the contractions, qvce sepissime inveniuntur in antiquis libris.* In the Metz MS. the contraction for the as is omitted. Several examples of reckoning time by this method occur in the classical writings, especially in Pliny, t as also in some MSS. of the eleventh and twelfth centuries in the British Museum. Bede wrote a tract on this fractional notation, and he adds, hcec ponderum vocabula, vel characteres non modo ad pecuniam mensurandam, verum ad qucevis corpora, sive tempora dimetienda, conveniunt. A MS. in the Public Library (Kk. v. 32) contains an explanation in • Euclidis Elementa ex Campano a Zamberto, fol. Par. 1516, p. 248. f Archaeologia, vol. xxvi. p. 159. Vid. Bedae Opera. Edit. Bas. 1563, t. i., col. 101, 14-1, et 182. MS. Arund. Mus. Brit. 25, f. 124, et 356, f. 45. APPENDIX. Ill Saxon, which very much coincides with that given by Bede, and would appear to be taken from it It would be impossible, with the few materials yet brought to light, to conjecture with any great probability how far these Boetian contractions may have influenced the introduction, or co-operated with the Arabic system to the formation, of our present numerical notation. It appears to me highly probable that the two systems became united, because the middle age forms of the figure five coincide with the Boetian mark for the same numeral, and those of two others are very similar. The idea of local position, again, may have had an indepen- dent European origin ; the inconveniences of the abacus on paper would have suggested it by destroying the distinguishing boundaries, and inventing an arbitrary hieroglyphic for the representation of an empty square. 112 APPENDIX. II. NOTES ON EAKLY ALMANACS.* The following short paper has been compiled from notes collected at various times, and without any intention of placing a dissertation on the subject before the public. I mention this merely to suggest to the reader that no connected history of almanacs has been attempted, and that it will be unfair to view what is here placed before him as any other than an attempt to abridge the labour of a suc- cessor who might wish, at some future period, to dive more deeply into the subject. The early history of almanacs is involved in much ob- scurity. The Egyptians, indeed, possessed instruments answering most of the same purposes : but the log calendars are the most ancient almanacs, properly so called. Ver- steganf derives their name from a Saxon origin, viz. al-mon-aght, or the observation of all the moons, that being the purpose for which they were originally made : an eastern origin would appear to me to be more probable. They are doubtless of high antiquity, and, if we can be guided by the errors of the more modern ones in their* * This has been printed in the "Companion to the British Almanac" for 1839; to the kindness of Mr. Knight, the publisher of that useful perio- dical, I am indebted for power to reprint it here. f Restitution of decayed intelligence, p. 64. APPENDIX. 113 ecclesiastical computation, we might refer them to the second or third century.* Gruter has delineated one at Rome, and which is said to have been used by the Goths and Vandals : this was cut in elm, though most are in box, and some few in fir, brass, horn, &c. Each of these calendars contains four sides, for the four quarters of the year, and gives the golden numbers, epacts, dominical letter, &c. The numerical notation is imperfect but curi- ous ; dots are put for the first four digits, a mark similar to the Roman numeral V, for five ; this mark, and additional dots for the next four, and the algebraical sign + for ten. Specimens of these logs may be seen in the British Museumf ; and, as they are not uncommon, it is unnecessary to enter into further detail. Before I commence with written almanacs, it will be neces- sary to remark the distinction between astronomical and ecclesiastical calendars, the first of which contain astrono- mical computations, and the other lists of saints' days, and other matters relative to the church ; sometimes, indeed, both are found united, although the latter claim a higher antiquity, being prefixed to most ancient Latin manuscripts of the Scriptures. The folding calendars were, perhaps, the most ancient forms of them, and merit particular attention. Several of these are in the British Museum,J and at Oxford ; one of them was written in the year 1430, and is in English ; but the writer confesses his inability to find suitable expressions * MS. Harl., Mus. Brit., 5958. f MSS. Harl., 197, 198. The last of these is a modern one used in Derbyshire or Staffordshire, and cut, probably, in the latter part of the 17th century : the other one is much earlier, though perhaps not of very high antiquity. Others may be seen in the Ashmolean Museum, and in St. John's College, Cambridge. I refer to Dr. Plott's History of Staffordshire for a very good description of them. (See also Brady's Clavis Calendaria.) % Cotton Rolls, viii. 26 ; MSS. Harl. 937, 3812, 5311 ; MSS. Sloan. 996, 2250 ; which last is the calendar of John Somers, afterwards mentioned. There is also one in the Ashmolean collection at Oxford in singularly fine preservation. 114 APPENDIX. for the technical terms which were derived for the most part from Arabic, for defawte of terms convenyent in our moder langage. In the Pepysian library at Cambridge there is one printed by Wynkin de Worde, in octo-decimo, which, in its original form, folds up from a small folio sheet of vellum ; it bears the date of 1523.* The standard almanacs emanated from Oxford,* the seat of British science throughout the middle ages: in fact, before Newton's time, Cambridge was a blank, and the only scientific names that cheer the pages of the history of its early literature are Holbroke of St. Peter's College, Buckley of King's, and Dee of St. John's : the first known by his astronomical tables, the second by a plagiarism of a method of extracting the roots of fractions from Robert Record, and the third a memorable instance of one of the greatest men of his time uniting the pure truths of science with the grossest absurdities. All three were astrologers,! owing, perhaps, more to the place of their education than to the individuals themselves. There has been some dispute relative to the authenticity of Roger Bacon's calendar, of which there is a MS. in the British Museum : the following is an exact transcript of the commencement : — " Kalendarium sequens extractum est a tabulis tholetanis . anno domini . 1292 . factus ad meridiem civitatis tholeti que in Hispania scita est cujus meridianus non multum distat a meridiano medii puncti Hibernie in quo . 3 . con- tinents." f. 2. If we retain factus, it cannot be translated, but, for- tunately, the other MS. at Oxford has factum, and this must evidently be the true reading. Professor Peacock * Hartshorne's ' Book Rarities of Cambridge.' f Holbroke is admitted by all to bave been an astrologer. Buckley wrote a treatise which involves the principles (MS. Bib. Reg. 12 A. xxv.) ; and with respect to Dr. Dee, no doubt can arise. APPENDIX. 115 writes factis, but there is not, as far as I know, any MS. authority for it. With respect to the author of it, the Bodleian MS., in a coeval rubric, states the calendar to have been written a fratre Rogero Baco?i ; while the Cotton MS., not having any original title, is ascribed to Roger Bacon, in a hand of the 1.7th century : both of the MSS. belong to the 14th century. In the Harleian collection (No. 941) is a MS. on the length of the days throughout the year, stated to have been made at Oxyn- forde be the new kalendere and proved in all the univer- sity : this new kalendere may possibly refer to Roger Bacon's ; but there are not sufficient data to enable us to attain an approach to certainty. The calendar of John Somers, of Oxford, written in 1380, was one of the most popular of the time: there is generally appended to it, Tabula docens algorismum legere, cujus utilitas est in brevi satis spatio numerum magnum comprehendere. Et quia numeri in kalendario positi vix excedunt sexaginta, ultra illam summam non est protensa* Several English translations of this tract are among the Ashmolean MSS. We have likewise in MS. Almanack Profacii Judei, which is very ancient. Walter de Elvendene wrote a calendar in 1327,f and Nicholas de Lynna published another in 1386.J Sometimes these calendars are found in rolls. In the library at Lambeth Palace is a very curious calendar in the English language, written in 1460; at the end is a table of eclipses from 1460 to 1481 ; but a very perfect volvelle is most worthy of notice, because those instruments are generally found imperfect. In the Cot- tonian collection is another English calendar, written about * MS. Bib. Cott. Mus. Brit. Vespas. E. vii. f. 4. f MS. Sloan. Mus. Brit. 286. J MS. Ashm. Oxon. 5. 116 APPENDIX. 1450, but so much damaged by the fire that the nature of it cannot be seen. In Trinity College, Cambridge, there is a MS., said to have been composed in 1347, and entitled, An Almanak, translated in perpetuite, out of Arabike into Latin ; and in the same library I find The Effe- merides of John of Mounte Riol, a German Prince of Astronomy ers. Professor Leslie mentions a very beautiful calendar in the library of the university of Edinburgh, with the date of 1 482 : he does not appear to be aware that they were common in MS. libraries, and he greatly overrates its value. There was printed at Hackney, in 1812, a small octavo volume, containing an account of an English almanac for the year 1386 : it contains a very large portion of astrono- mical and medical matter, but appears to be of little interest, save that it is the earliest one in English I have ever heard of. The contents of this calendar are as follow : — 1 . The houses of the planets and their properties. 2. The exposition of the signs. 3. Chronicle of events from the birth of Cain. In 1325 there was a grete hungur in England ; in 1333 a great tempest ; in 1349 the first, in 1361 the second, and in 1369 the third pestilence. It is curious to remark the clumsy method of expressing numbers consisting of more than two figures: for instance, we have 52mcc20 put for 52,220. This shows that the Arabic notation was even then but imperfectly understood among the common people. 4. To find the prime numbers. 5. Short notes on medicine. 6. On blood-letting. 7. A description of the table of the signs, and moveable feasts. 8. Quantitates diei artificialis. APPENDIX. 117 The extracts from this calendar are wretchedly trans- cribed, and evidently by one who was totally unacquainted with MSS. The clock or albion of Richard de Walingford, of St. Alban's, answered the purpose of a calendar. * This clock made, says Bale, who appears to have seen it, magno labore, majore sumptu, arte vero maxima, was considered the greatest curiosity of its time. In his account of it, which still remains in manuscript, we have the following definitions : — Albion est geometricum instrumentum : al- manac autem arismetricum. Peter Lightfoot's celebrated astronomical clock at Glastonbury may have been something of the same sort. Peter de Dacia, about 1300, published a calendar, of which there is a very early MS. in the Savilian library at Oxford : the condiciones planetarum are thus stated — Jupiter atque Venus boni, Saturnusque malignus ; Sol et Mercurius cum Luna sunt mediocres. The homo signorum, so common in later calendars, probably originated with him. The earliest almanac printed in England was the Sheape- hearcFs Kalender, translated from the French, and printed by Richard Pynson, in 1497. It contains a vast portion of extraneous matter. The following verses on the planets will, at the same time, give a good idea of the nature • In one almanac of the commencement of the 17th century (MS. Harl. 5937. Bagford Collect., s. 139) is a very singular method for finding the hour of the day, if in the country and without any watch. I refer to it merely out of charity to those right-hearted enthusiastic antiquaries who do not stick at trifles in pursuing researches that can in any way illustrate the customs of our ancestors in the good old times. To those who only value the researches of antiquaries in proportion as they are likely to furnish some evident tan- gible utility, I would willingly spare their time in recurring to a method which, though it might excite their ridicule, could never, from its nature, be brought into practice in the present age. 118 APPENDIX. of the astrological information in this and other calendars of the period : — " Some hot, some colde, some moyst, some dry, If three be good, foure be worse at the most. Saturn e is hyest and coldest, being full old, And Mars, with his bluddy swerde, ever ready to kyll ; Jupiter very good, and Venus maketh lovers glad, Sol and Luna is half good and half ill, Mercury is good and will verily And hereafter shalt thou know; Whiche of the seven most worthy be, And who reigneth hye, and who a lowe ; Of every planets propertie, Which is the best among them all, That causeth welth, sorrowe, or sinne, Tarry and heare sone thou shalt, Speake softe, for now I beginne." Aiterwards follow some prognostications of the weather. The following method to knowe what wether shall be all the yere after the chaunge of every moone by the prime dayes, is taken from a MS. in Lambeth Palace : — " Sondaye pry me, drye wether. Mondaye pryme, moyst wether. Teusdaye pryme, cold and wynde. Wenesdaye pryme, mervelous. Thursdaye pryme, sonne and clere. Frydaye pryme, fayre and fowle. Saturdaye pryme, rayne." Prognostications of the weather were early matters of reproach — " Astronomyers also aren at ere whittes ende,. Of that was calculed of the clymat the contrye thei fyndeth. And in Heber's library was a little tract of three leaves, entitled fA Mery Prognostication '— " For the yere of Chryste's incamacyon, A thousande fyve hundreth fortye and foure. This to prognosticate I may be bolde, That whan the new yere is come, gone is the olde." APPENDIX. 119 Henry VI II. issued a proclamation against such false prognostications as this tract was intended to ridicule, but still no printer ventured to put his name to it. Not long after to believe them was a crime ; " as for astro- logicall and other like vaine predictions or abodes," says Thomas Lydiat, "I thanke God I was never addicted to them." * Johannes de Monte-Regio, in 1472, composed the earliest European almanac that issued from the press ; and, before the end of that century, they became com- mon on the Continent. In England they were not in general use until the middle of the sixteenth century. Most of the best mathematicians of the time were em- ployed in constructing them ,• but, before the end of the following century, almanac-makers began to form a distinct body, and, though they often styled themselves " studentes in the artes mathematicall," very few of them were at all celebrated in the pure sciences. It may not be wholly irrelevant here to make some few observations on the memory-rhymes found in some almanacs of the present day, and which date their origin to a much earlier period. The well-known lines, used by many for recalling to their recollection the number of days in each month, I find in Winter's Cambridge Alma- nac for 1635, under the following slightly-varied form — " Aprill, June, and September, Thirty daies have as November ; Ech month else doth never vary From thirty-one, save February ; Wich twenty-eight doth still confine, Save on Leap-yeare, then twenty-nine." And the nursery-rhymes, commencing " Multiplication is my vexation," were certainly made before 1570.f * MS. Bodl. 662. t Professor Davies's Key to Hutton's Mathematics, p. 17. 120 APPENDIX. The early history of ecclesiastical computation is inti- mately connected with that of calendars. Dionysius Exiguus was one of the first who wrote on the subject : after him, Bede, Gerlandus, Alexander de Villa Dei, and Johannes de Sacro-Bosco, were the most celebrated. The Massa Compoti of Alexander de Villa Dei, so common in MS., is perhaps the most singular tract on the subject that has come down to us : his reason for the title of the book is exceedingly curious : — Sicut de multis laminis ceris in conflatorio massa una efficitur, ideo librum istum vocari volui massam compoti. I cannot conclude without mentioning the ' Almanac and Prognostication' of Leonard Digges, which was so often reprinted in the latter half of the sixteenth century: it is filled with the most extravagant astrological absur- dities, and a table of weather predictions. With respect to the latter, however, I have had the curiosity to test its accuracy for some months in comparison with our two celebrated weather almanacs, and, on the average, have found it to be quite as " neare the marke " as either of them. THE END. CAMBRIDGE : PRINTED BY METCALFE AND PALMER, TRINITY STREET. Hlf mi HOME USE CIRCULATION DEPARTMENT MAIN LIBRARY This book is due on the last date stamped below. 1 -month loans may be renewed by calling 642-3405. 6-month loans may be recharged by bringing books to Circulation Desk. Renewals and recharges may be made 4 days prior to due date. ALL BOOKS ARE SUBJECT TO RECALL 7 DAYS AFTER DATE CHECKED OUT. 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