IMPRINTEDat London in Pouleschurcheyarde, at the signe of the Bra-senserpent, by Reynold Wolfe.Cum priuilegio ad imprimen-dum solum.ANNO DOMINI.M.D.L.I.brazen serpent with text NVM. XXI.Title page:The pathway toKNOWLEDG,CONTAI-NING THE FIRST PRIN-ciples of Geometrie, as theymay moste aptly be applied vn-to practice, bothe for vse ofinstrumentes Geome-tricall, and astrono-micall andalso for proiection of plattes in eueryekinde, and therfore much ne-cessary for all sortes ofmen.Dedication:TO THE MOST NO-ble and puissaunt prince Edwarde thesixte by the grace of God, of En-gland Fraunce and Ireland kynge, de-fendour of the faithe, and of theChurche of England and Ire-lande in earth the su-preme head.Title page of Second Book:THE SECOND BOOKEOF THE PRINCIPLESof Geometry, containing certaineTheoremes, whiche may be cal-led Approued truthes. And be asit were the moste certainegroundes, wheron thepractike cõclusionsof Geometry arfounded.[Leaf]Whervnto are annexed certaine declarations byexamples, for the right vnderstanding of thesame, to the ende that the simple readermight not iustly cõplain of hardnesor obscuritee, and for the samecause ar the demonstra-tions and iust profesomitted, vntill amoreconueni-enttime.1551.“Theorems of Geometry”:The Theoremes of Geometry, beforeWHICHE ARE SET FORTHEcertaine grauntable requesteswhich serue for demonstrationsMathematicall.Transcriber’s NotesTerminologyCorrectionsIllustrationsNotesGreekLanguage and TypographyThis information is for readers who know more about geometry than about mid-16th-century English.The lettersuandvfollow the conventional “initial v, non-initial u” pattern except in numbers (xv, iv). The lower-casejform occurs only as the last digit of a number (ij, xxj); upper-caseIandJshare a form, always read as I. Italic doubleswas printed as an ſ+s ligature, similar to the German ß; it is shown as simple “ss”. Capital and lower-casewwere often used interchangeably. Words split across line breaks may or may not have a hyphen.The word “other” is used interchangeably with both “or” and “either”; similarly, “nother” is used in place of “nor” and “neither”. The expression “an other” is almost always written as two words.The spellings “then(ne)” and “than(ne)” are used interchangeably; “than(ne)” is rare. The spelling “liyng”, both by itself and as the end of a longer word, is used consistently.TerminologyExplanations of these terms are scattered through the book. They are grouped here for convenience.right linestraight linegemow (line)parallelgemew = twinsquare(squire, squyre)quadrilateralalso means angle square as described under hexagons (“siseangles”)likeiammeparallelogramiam(me) = jamb = limb, sidelongsquarerectangletouch linetangentcantlesegment of a circlecantle = sliceCorrectionsUnless otherwise noted, spelling, punctuation and capitalization are unchanged. Forms were regularized only where there was a very large disparity between the expected form and the apparent errors (for example, a thousand “A.B” against a dozen “A,B”), or a flagrant misprint such as “cnt” for “cut”.Number forms such as “those. ij. last” or “line. A.B.” were silently regularized to “those .ij. last” and “line .A.B.” Missing sentence-final periods at the end of a printed line were silently supplied.Unusual forms or combinations—especially phrase breaks where a comma is followed by a capital letter, or a period by a lower-case letter—are unchanged but noted withpopups.IllustrationsA number of illustrations contain errors such as unmarked or mislabeled points (“circle B.C.D” where only C and D are labeled). Labels added by the transcriber are shown in grey; moved or transposed labels are shown in red, withmouse-hover explanation.Notes1)if they can with their wysedome ouercome all vyces. Of the firste of those three sortesText reads “... their rwysedome ... / ... those th ee sortes ...” on consecutive lines:partial page image2)Pagination as shown by signature numbers demands another leaf (two pages) between the end of the Preface and the beginning of the body text. But no text is missing, and the facsimile has no blank pages.3)and yet the ij. lesser sides togither ar greater then it.Text reads “... yet thr / ... ar greate” at consecutive line-ends:partial page imageGreekAll Greek is shown as printed. Errors or anomalies include missing, misplaced or incorrect diacritics; the letterform σ for ς; and word-final μ (mu) for ν (nu).ἐίπερ γὰρ ἀδικεῖμ χρὴ, τυραννΐδος περῒ κάλλιστομ ἀδικεῖμ, τ’ ἄλλα δ’ ἐυσεβεῖμ χρεῶμeiper gar adikeim chrê, turannidos peri kallistom adikeim, t’ alla d’ eusebeim chreôm.Φΐλιππος Αριστοτέλει χαίρειμ.ἔσθι μοι γεγονότα ὑομ. πολλὴμ οὖμ τοῖσ θεοῖσ χάριμ ἔχω, ὀυχ ὅυτωσ ἐπῒ τῆ γεννήσει του παιδόσ, ὡσ ἐπῒ τῷ κατὰ τὴμ σὴμ ἡλικῒαμ αὐτόμ γεγονέναι ἐλπΐζω γὰρ αὐτὸμ ὑπὸ σοῦ γραφέντα καὶ παιδευθέντα ἄξιομ ἔσεσθαι καὶ ἑιμῶμ καὶ τῆς τῶμ τραγμάτωμ διαδοχῆσ.Philippos Aristotelei chaireim.esthi moi gegonota huom. pollêm oum tois theois charim echô, ouch houtôs epi tê gennêsei tou paidos, hôs epi tô kata têm sêm hêlikiam autom gegonenai elpizô gar autom hupo sou graphenta kai paideuthenta axiom esesthai kai heimôm kai tês tôm pragmatôm diadochês.Ἄλέζανδρος Αρισοτέλει εὖ πράττειμ.Ὂυκ ὀρθῶσ ἐπόιησασ ἐκδοὺσ τοὺσ ἀκροαματικόυσ τῶμ λόγωμ, τΐνι γὰρ διοισομην ἡμεῖσ τῶμ ἄλλωμ, ἐι καθ’ οὕσ ἐπαιδεύθημεν λόγουσ, ὅυτοι πάντωμ ἔσονταιν κοινόι, ἐγὼ δὲ βουλοί μημ ἅμ ταῖσ περι τὰ ἄριστα ἐμπειρΐαισ, ἢ τὰισ δυνάμεσι διαφέριμ. ἔρρωσο.Alezandros Arisotelei eu pratteim.Ouk orthôs epoiêsas ekdous tous akroamatikous tôm logôm, tini gar dioisomên hêmeis tôm allôm, ei kath’ hous epaideuthêmen logous, houtoi pantôm esontai koinoi, egô de bouloi mêm am tais peri ta arista empeiriais, ê tais dunamesi diapherim. errôso.
brazen serpent with text NVM. XXI.
The pathway toKNOWLEDG,CONTAI-NING THE FIRST PRIN-ciples of Geometrie, as theymay moste aptly be applied vn-to practice, bothe for vse ofinstrumentes Geome-tricall, and astrono-micall andalso for proiection of plattes in eueryekinde, and therfore much ne-cessary for all sortes ofmen.
TO THE MOST NO-ble and puissaunt prince Edwarde thesixte by the grace of God, of En-gland Fraunce and Ireland kynge, de-fendour of the faithe, and of theChurche of England and Ire-lande in earth the su-preme head.
THE SECOND BOOKEOF THE PRINCIPLESof Geometry, containing certaineTheoremes, whiche may be cal-led Approued truthes. And be asit were the moste certainegroundes, wheron thepractike cõclusionsof Geometry arfounded.
[Leaf]
Whervnto are annexed certaine declarations byexamples, for the right vnderstanding of thesame, to the ende that the simple readermight not iustly cõplain of hardnesor obscuritee, and for the samecause ar the demonstra-tions and iust profesomitted, vntill amoreconueni-enttime.
1551.
The Theoremes of Geometry, beforeWHICHE ARE SET FORTHEcertaine grauntable requesteswhich serue for demonstrationsMathematicall.
Transcriber’s NotesTerminologyCorrectionsIllustrationsNotesGreekLanguage and TypographyThis information is for readers who know more about geometry than about mid-16th-century English.The lettersuandvfollow the conventional “initial v, non-initial u” pattern except in numbers (xv, iv). The lower-casejform occurs only as the last digit of a number (ij, xxj); upper-caseIandJshare a form, always read as I. Italic doubleswas printed as an ſ+s ligature, similar to the German ß; it is shown as simple “ss”. Capital and lower-casewwere often used interchangeably. Words split across line breaks may or may not have a hyphen.The word “other” is used interchangeably with both “or” and “either”; similarly, “nother” is used in place of “nor” and “neither”. The expression “an other” is almost always written as two words.The spellings “then(ne)” and “than(ne)” are used interchangeably; “than(ne)” is rare. The spelling “liyng”, both by itself and as the end of a longer word, is used consistently.TerminologyExplanations of these terms are scattered through the book. They are grouped here for convenience.right linestraight linegemow (line)parallelgemew = twinsquare(squire, squyre)quadrilateralalso means angle square as described under hexagons (“siseangles”)likeiammeparallelogramiam(me) = jamb = limb, sidelongsquarerectangletouch linetangentcantlesegment of a circlecantle = sliceCorrectionsUnless otherwise noted, spelling, punctuation and capitalization are unchanged. Forms were regularized only where there was a very large disparity between the expected form and the apparent errors (for example, a thousand “A.B” against a dozen “A,B”), or a flagrant misprint such as “cnt” for “cut”.Number forms such as “those. ij. last” or “line. A.B.” were silently regularized to “those .ij. last” and “line .A.B.” Missing sentence-final periods at the end of a printed line were silently supplied.Unusual forms or combinations—especially phrase breaks where a comma is followed by a capital letter, or a period by a lower-case letter—are unchanged but noted withpopups.IllustrationsA number of illustrations contain errors such as unmarked or mislabeled points (“circle B.C.D” where only C and D are labeled). Labels added by the transcriber are shown in grey; moved or transposed labels are shown in red, withmouse-hover explanation.Notes1)if they can with their wysedome ouercome all vyces. Of the firste of those three sortesText reads “... their rwysedome ... / ... those th ee sortes ...” on consecutive lines:partial page image2)Pagination as shown by signature numbers demands another leaf (two pages) between the end of the Preface and the beginning of the body text. But no text is missing, and the facsimile has no blank pages.3)and yet the ij. lesser sides togither ar greater then it.Text reads “... yet thr / ... ar greate” at consecutive line-ends:partial page imageGreekAll Greek is shown as printed. Errors or anomalies include missing, misplaced or incorrect diacritics; the letterform σ for ς; and word-final μ (mu) for ν (nu).ἐίπερ γὰρ ἀδικεῖμ χρὴ, τυραννΐδος περῒ κάλλιστομ ἀδικεῖμ, τ’ ἄλλα δ’ ἐυσεβεῖμ χρεῶμeiper gar adikeim chrê, turannidos peri kallistom adikeim, t’ alla d’ eusebeim chreôm.Φΐλιππος Αριστοτέλει χαίρειμ.ἔσθι μοι γεγονότα ὑομ. πολλὴμ οὖμ τοῖσ θεοῖσ χάριμ ἔχω, ὀυχ ὅυτωσ ἐπῒ τῆ γεννήσει του παιδόσ, ὡσ ἐπῒ τῷ κατὰ τὴμ σὴμ ἡλικῒαμ αὐτόμ γεγονέναι ἐλπΐζω γὰρ αὐτὸμ ὑπὸ σοῦ γραφέντα καὶ παιδευθέντα ἄξιομ ἔσεσθαι καὶ ἑιμῶμ καὶ τῆς τῶμ τραγμάτωμ διαδοχῆσ.Philippos Aristotelei chaireim.esthi moi gegonota huom. pollêm oum tois theois charim echô, ouch houtôs epi tê gennêsei tou paidos, hôs epi tô kata têm sêm hêlikiam autom gegonenai elpizô gar autom hupo sou graphenta kai paideuthenta axiom esesthai kai heimôm kai tês tôm pragmatôm diadochês.Ἄλέζανδρος Αρισοτέλει εὖ πράττειμ.Ὂυκ ὀρθῶσ ἐπόιησασ ἐκδοὺσ τοὺσ ἀκροαματικόυσ τῶμ λόγωμ, τΐνι γὰρ διοισομην ἡμεῖσ τῶμ ἄλλωμ, ἐι καθ’ οὕσ ἐπαιδεύθημεν λόγουσ, ὅυτοι πάντωμ ἔσονταιν κοινόι, ἐγὼ δὲ βουλοί μημ ἅμ ταῖσ περι τὰ ἄριστα ἐμπειρΐαισ, ἢ τὰισ δυνάμεσι διαφέριμ. ἔρρωσο.Alezandros Arisotelei eu pratteim.Ouk orthôs epoiêsas ekdous tous akroamatikous tôm logôm, tini gar dioisomên hêmeis tôm allôm, ei kath’ hous epaideuthêmen logous, houtoi pantôm esontai koinoi, egô de bouloi mêm am tais peri ta arista empeiriais, ê tais dunamesi diapherim. errôso.
TerminologyCorrectionsIllustrationsNotesGreek
This information is for readers who know more about geometry than about mid-16th-century English.
The lettersuandvfollow the conventional “initial v, non-initial u” pattern except in numbers (xv, iv). The lower-casejform occurs only as the last digit of a number (ij, xxj); upper-caseIandJshare a form, always read as I. Italic doubleswas printed as an ſ+s ligature, similar to the German ß; it is shown as simple “ss”. Capital and lower-casewwere often used interchangeably. Words split across line breaks may or may not have a hyphen.
The word “other” is used interchangeably with both “or” and “either”; similarly, “nother” is used in place of “nor” and “neither”. The expression “an other” is almost always written as two words.
The spellings “then(ne)” and “than(ne)” are used interchangeably; “than(ne)” is rare. The spelling “liyng”, both by itself and as the end of a longer word, is used consistently.
Explanations of these terms are scattered through the book. They are grouped here for convenience.
parallelgemew = twin
square(squire, squyre)
quadrilateralalso means angle square as described under hexagons (“siseangles”)
parallelogramiam(me) = jamb = limb, side
segment of a circlecantle = slice
Unless otherwise noted, spelling, punctuation and capitalization are unchanged. Forms were regularized only where there was a very large disparity between the expected form and the apparent errors (for example, a thousand “A.B” against a dozen “A,B”), or a flagrant misprint such as “cnt” for “cut”.
Number forms such as “those. ij. last” or “line. A.B.” were silently regularized to “those .ij. last” and “line .A.B.” Missing sentence-final periods at the end of a printed line were silently supplied.
Unusual forms or combinations—especially phrase breaks where a comma is followed by a capital letter, or a period by a lower-case letter—are unchanged but noted withpopups.
A number of illustrations contain errors such as unmarked or mislabeled points (“circle B.C.D” where only C and D are labeled). Labels added by the transcriber are shown in grey; moved or transposed labels are shown in red, withmouse-hover explanation.
1)if they can with their wysedome ouercome all vyces. Of the firste of those three sortes
Text reads “... their rwysedome ... / ... those th ee sortes ...” on consecutive lines:
partial page image
2)Pagination as shown by signature numbers demands another leaf (two pages) between the end of the Preface and the beginning of the body text. But no text is missing, and the facsimile has no blank pages.
3)and yet the ij. lesser sides togither ar greater then it.
Text reads “... yet thr / ... ar greate” at consecutive line-ends:
partial page image
All Greek is shown as printed. Errors or anomalies include missing, misplaced or incorrect diacritics; the letterform σ for ς; and word-final μ (mu) for ν (nu).
ἐίπερ γὰρ ἀδικεῖμ χρὴ, τυραννΐδος περῒ κάλλιστομ ἀδικεῖμ, τ’ ἄλλα δ’ ἐυσεβεῖμ χρεῶμeiper gar adikeim chrê, turannidos peri kallistom adikeim, t’ alla d’ eusebeim chreôm.Φΐλιππος Αριστοτέλει χαίρειμ.ἔσθι μοι γεγονότα ὑομ. πολλὴμ οὖμ τοῖσ θεοῖσ χάριμ ἔχω, ὀυχ ὅυτωσ ἐπῒ τῆ γεννήσει του παιδόσ, ὡσ ἐπῒ τῷ κατὰ τὴμ σὴμ ἡλικῒαμ αὐτόμ γεγονέναι ἐλπΐζω γὰρ αὐτὸμ ὑπὸ σοῦ γραφέντα καὶ παιδευθέντα ἄξιομ ἔσεσθαι καὶ ἑιμῶμ καὶ τῆς τῶμ τραγμάτωμ διαδοχῆσ.Philippos Aristotelei chaireim.esthi moi gegonota huom. pollêm oum tois theois charim echô, ouch houtôs epi tê gennêsei tou paidos, hôs epi tô kata têm sêm hêlikiam autom gegonenai elpizô gar autom hupo sou graphenta kai paideuthenta axiom esesthai kai heimôm kai tês tôm pragmatôm diadochês.Ἄλέζανδρος Αρισοτέλει εὖ πράττειμ.Ὂυκ ὀρθῶσ ἐπόιησασ ἐκδοὺσ τοὺσ ἀκροαματικόυσ τῶμ λόγωμ, τΐνι γὰρ διοισομην ἡμεῖσ τῶμ ἄλλωμ, ἐι καθ’ οὕσ ἐπαιδεύθημεν λόγουσ, ὅυτοι πάντωμ ἔσονταιν κοινόι, ἐγὼ δὲ βουλοί μημ ἅμ ταῖσ περι τὰ ἄριστα ἐμπειρΐαισ, ἢ τὰισ δυνάμεσι διαφέριμ. ἔρρωσο.Alezandros Arisotelei eu pratteim.Ouk orthôs epoiêsas ekdous tous akroamatikous tôm logôm, tini gar dioisomên hêmeis tôm allôm, ei kath’ hous epaideuthêmen logous, houtoi pantôm esontai koinoi, egô de bouloi mêm am tais peri ta arista empeiriais, ê tais dunamesi diapherim. errôso.
ἐίπερ γὰρ ἀδικεῖμ χρὴ, τυραννΐδος περῒ κάλλιστομ ἀδικεῖμ, τ’ ἄλλα δ’ ἐυσεβεῖμ χρεῶμ
eiper gar adikeim chrê, turannidos peri kallistom adikeim, t’ alla d’ eusebeim chreôm.
Φΐλιππος Αριστοτέλει χαίρειμ.
ἔσθι μοι γεγονότα ὑομ. πολλὴμ οὖμ τοῖσ θεοῖσ χάριμ ἔχω, ὀυχ ὅυτωσ ἐπῒ τῆ γεννήσει του παιδόσ, ὡσ ἐπῒ τῷ κατὰ τὴμ σὴμ ἡλικῒαμ αὐτόμ γεγονέναι ἐλπΐζω γὰρ αὐτὸμ ὑπὸ σοῦ γραφέντα καὶ παιδευθέντα ἄξιομ ἔσεσθαι καὶ ἑιμῶμ καὶ τῆς τῶμ τραγμάτωμ διαδοχῆσ.
Philippos Aristotelei chaireim.
esthi moi gegonota huom. pollêm oum tois theois charim echô, ouch houtôs epi tê gennêsei tou paidos, hôs epi tô kata têm sêm hêlikiam autom gegonenai elpizô gar autom hupo sou graphenta kai paideuthenta axiom esesthai kai heimôm kai tês tôm pragmatôm diadochês.
Ἄλέζανδρος Αρισοτέλει εὖ πράττειμ.
Ὂυκ ὀρθῶσ ἐπόιησασ ἐκδοὺσ τοὺσ ἀκροαματικόυσ τῶμ λόγωμ, τΐνι γὰρ διοισομην ἡμεῖσ τῶμ ἄλλωμ, ἐι καθ’ οὕσ ἐπαιδεύθημεν λόγουσ, ὅυτοι πάντωμ ἔσονταιν κοινόι, ἐγὼ δὲ βουλοί μημ ἅμ ταῖσ περι τὰ ἄριστα ἐμπειρΐαισ, ἢ τὰισ δυνάμεσι διαφέριμ. ἔρρωσο.
Alezandros Arisotelei eu pratteim.
Ouk orthôs epoiêsas ekdous tous akroamatikous tôm logôm, tini gar dioisomên hêmeis tôm allôm, ei kath’ hous epaideuthêmen logous, houtoi pantôm esontai koinoi, egô de bouloi mêm am tais peri ta arista empeiriais, ê tais dunamesi diapherim. errôso.