Angvli eorvmqve mensvra: gradvs et radiani

Svmmarivm

Haec lectio angvlos orientatos definit, qvi conversionem cvm svo sensv metivntvr et integram conversionem excedere possvnt, eorvmqve dvas mensvras vsitatas: gradvm sexagesimalem, antiqva conventione constitvtvm, et radianvm, qvi est longitvdo arcvs qvem angvlvs in circvmferentia vnitaria svbtendit. Admittit longitvdinem circvmferentiae, proportionem arcvvm ad radios et conservationem longitvdinvm per reflexiones, nvmervm $\pi$ definit vt dimidiam longitvdinem circvmferentiae vnitariae et axioma percvrsvs per longitvdinem arcvs envntiat, qvod cviqve nvmero reali pvnctvm illivs circvmferentiae attribvit; inde dedvcit qvomodo percvrsvs se habeat in altera sva dimidia conversione, regvlam conversionis inter gradvs et radianos, longitvdinem arcvs, aream sectoris et theorema angvli inscripti. Deinde angvlos in positione normali collocat, demonstrat omnem angvlvm vno tantvm modo ad angvlvm primae conversionis redigi et vnam conversionem non bis per idem pvnctvm transire, angvlos coterminales describit et velocitatem angvlarem cvm lineari conivngit. Problemata Terram vt Eratosthenes metivntvr, segmentvm circvli compvtant, conversionem per catenam transmittvnt, aream sectoris perimetri fixi maximam reddvnt et nvmerant qvotiens indices horologii inter se congrvant.

Proposita discendi

Latvs initiale et latvs terminale angvli orientati distingvere, evm gradibvs, minvtis et secvndis metiri, et signvm eivs ex sensv conversionis agnoscere.
Axioma percvrsvs per longitvdinem arcvs envntiare, eo radianvm definire et mensvras inter gradvs et radianos convertere, praesertim angvlorvm insignivm.
Formvlas longitvdinis arcvs et areae sectoris, theorema angvli inscripti et relationem inter velocitatem angvlarem et linearem demonstrare et adhibere.
Qvemlibet angvlvm ad angvlvm primae conversionis redigere, exsistentiam et vnicitatem redvctionis demonstrando, et divdicare vtrvm dvo angvli coterminales sint.

Angvli orientati

In geometria elementari angvlvs est figvra dvabvs semirectis commvnem originem habentibvs formata, eivsqve mensvra inter $0^\circ$ et $180^\circ$ iacet. Ad conversiones investigandas, vt rotae, vt indicvm horologii, vt Terrae circa axem svvm, haec notio non svfficit: refert qvantvm et qvo sensv conversvm sit, et conversio plvres orbes conficere potest. Qvam ob rem angvlvs habetvr vt conversionis effectvs.

Angvlvs orientatvs determinatvr semirecta, qvae latvs initiale appellatvr, circa originem svam, qvae vertex appellatvr, conversa donec positionem finalem occvpet, qvae latvs terminale appellatvr, vna cvm conversione peracta. Angvlvs est positivvs si conversio fit sensv contrario motvi indicvm horologii (contra motvm horologii), negativvs si fit eodem sensv qvo indices horologii moventvr (secvndvm motvm horologii). Angvlvs nvllvs est is qvi non convertitvr.

Idem par latervm mvltis angvlis orientatis respondere potest: latvs terminale eandem positionem occvpat sive qvarta conversionis portione contra motvm horologii convertitvr, sive tribvs qvartis secvndvm motvm horologii, sive qvarta portione et insvper integra conversione. Hi angvli diversi svnt, qvia conversiones diversas describvnt, qvamvis in eodem latere desinant; tertia sectio definit qvomodo inter se cohaereant.

Systema sexagesimale

Antiqvissima angvlorvm mensvra integram conversionem in $360$ arcvs aeqvales dividit, et gradvm appellat angvlvm qvi vnvmqvemqve eorvm svbtendit: angvlvs $g$ gradvs metitvr si arcvs eivs $g$ vicibvs arcvm vnivs gradvs continet, ita vt mensvra in gradibvs longitvdini arcvs proportionalis sit. Vnvsqvisqve gradvs in $60$ minvta dividitvr, et vnvmqvodqve minvtvm in $60$ secvnda. Systema ab astronomia Babylonia oritvr, qvae basi sexaginta nvmerabat, vt in lectione cvrsvs I de systematibvs nvmerationis positionalis explicatvr; nvmervs $360$ id commodi habet qvod dividi potest per $2$ , $3$ , $4$ , $5$ , $6$ , $8$ , $9$ , $10$ et $12$ , inter alios. Angvlvs rectvs $90^\circ$ metitvr, angvlvs planvs $180^\circ$ , integra conversio $360^\circ$ ; angvlvs negativvs cvm signo minvs metitvr, ita vt qvarta conversionis portio secvndvm motvm horologii $-90^\circ$ metiatvr. Angvlvs 40 gradvvm, 15 minvtorvm et 36 secvndorvm, qvi breviter scribitvr post vnvmqvemqve nvmervm signo gradvs, vno apostropho pro minvtis et dvobvs pro secvndis, in gradibvs metitvr qvod infra scriptvm est, id est 40,26 gradvs:

40 + \frac{15}{60} + \frac{36}{3600} = 40 + \frac{1}{4} + \frac{1}{100} = \frac{2013}{50}

Nvmervs $360$ tamen conventio est: nihil in geometria circvli evm ab alio nvmero distingvit. Formvlae longitvdinis arcvs et areae sectoris, et postea fvnctionvm trigonometricarvm, simpliciores fivnt si angvli eo modo metivntvr qvi ex ipsa circvmferentia nascitvr.

Radianvs

Consilivm est angvlvm metiri longitvdine arcvs qvem svbtendit in circvmferentia radii $1$ cvivs centrvm est vertex eivs. Hoc postvlat vt sciatvr qvid sit longitvdo arcvs, et hic difficvltas apparet: arcvs non est vnio segmentorvm, eivsqve longitvdo definitvr vt valor ad qvem accedvnt longitvdines linearvm polygonarvm ei inscriptarvm, cvm latera earvm magis magisqve breviantvr, id est per limitem. Eivs investigatio ad cvrsvm Calcvlvs II pertinet, vbi longitvdines cvrvarvm et areae regionvm cvrvis terminatarvm compvtantvr. Hic facta qvae seqvvntvr admittvntvr.

Factvm admissvm (longitvdines et areae in circvlo). Omnis arcvs circvmferentiae longitvdinem habet, et omnis regio segmentis et arcvbvs terminata aream habet. Figvrae congrventes eandem longitvdinem vel eandem aream habent; si figvra in dvas dividitvr qvae non sibi svperponvntvr, mensvra eivs est svmma mensvrarvm membrorvm; et si regio aliam continet, area eivs non est minor, et maior est si differentia aliqvem discvm continet. Praeterea, si dvae circvmferentiae radios $r$ et $R$ habent, arcvs qvos idem angvlvs ad centrvm in eis svbtendit radiis proportionales svnt, et area disci radii $r$ est $\pi r^2$ .

Hoc factvm sine demonstratione admittitvr, qvia longitvdines arcvvm et areae regionvm cvrvarvm per limites definivntvr et integrali compvtantvr, in cvrsv Calcvlvs II. Postrema eivs clavsvla nvmervm $\pi$ notvm svpponit, qvi nvnc definitvr: nvmervs $\pi$ est dimidia longitvdo circvmferentiae vnitariae, id est circvmferentiae radii $1$ . Ex proportione arcvvm ad radios, angvlo integrae conversionis adhibita, circvmferentia radii $r$ longitvdinem $2\pi r$ habet, et qvotiens longitvdinis cvivslibet circvmferentiae et diametri eivs semper est $\pi$ . Admittitvr etiam $\pi$ irrationalem esse, eivsqve expressionem decimalem a 3,14159 incipere; praesertim $3 < \pi < 4$ .

Ad angvlos metiendos opvs est praeterea circvmferentiam per longitvdinem arcvs percvrrere: a pvncto proficisci et in ea longitvdinem datam progredi, in altervtrvm sensvm, et scire ad qvod pvnctvm perveniatvr. Omnia qvae haec pars de angvlis et fvnctionibvs trigonometricis dicit in hoc percvrsv nitvntvr, ita vt expediat accvrate envntiare qvid de eo admittatvr. Envntiatvr in plano Cartesiano, pro circvmferentia vnitaria $x^2 + y^2 = 1$ et eivs pvncto $A = (1, 0)$ ; cvm longitvdines non mvtentvr figvris motis, aeqve valet pro qvavis circvmferentia radii $1$ et qvovis pvncto initiali.

Axioma (percvrsvs per longitvdinem arcvs). Pro qvoqve nvmero reali $s$ exstat vnicvm pvnctvm $P(s)$ circvmferentiae vnitariae ad qvod pervenitvr proficiscendo a $A = (1, 0)$ et in ea longitvdinem $\abs{s}$ percvrrendo, contra motvm horologii si $s > 0$ et secvndvm motvm horologii si $s < 0$ , tot orbes conficiendo qvot opvs sit; pro $s = 0$ est $P(0) = A$ . Hic percvrsvs qvattvor proprietates habet qvae infra envmerantvr.

Conivnctio: si a pvncto $P(s)$ longitvdo $t \geq 0$ contra motvm horologii percvrritvr, pervenitvr ad $P(s + t)$ ; si secvndvm motvm horologii percvrritvr, pervenitvr ad $P(s - t)$ .
Periodicitas: integra conversio, in vtrvmvis sensvm, longitvdinem $2\pi$ circvmferentiae percvrrit et ad pvnctvm initiale redvcit; id est, $P(s + 2\pi) = P(s)$ pro omni nvmero reali $s$ .
Dimidia conversio: percvrsvs contra motvm horologii primvm per semicircvlvm svperiorem transit, a dextra ad sinistram. Accvrate, $P(\pi) = (-1, 0)$ , et si $0 \leq s < t \leq \pi$ , pvncta $P(s)$ et $P(t)$ ordinatam non negativam habent et abscissa ipsivs $P(s)$ maior est qvam ipsivs $P(t)$ .
Operimentvm: omne pvnctvm circvmferentiae vnitariae est $P(s)$ pro aliqvo nvmero reali $s$ .

Axioma sine demonstratione admittitvr eadem de cavsa ac factvm svpra envntiatvm: longitvdo arcvs per limitem definitvr, eivsqve proprietates in cvrsv Calcvlvs II demonstrantvr; operimentvm, qvod affirmat circvmferentiam percvrrendo nvllvm pvnctvm praeteriri, praeterea nititvr proprietate valoris intermedii fvnctionvm continvarvm, qvae in cvrsv Calcvlvs I demonstratvr. Proprietas dimidiae conversionis simplicem lectionem geometricam habet. Si $(x, y)$ in circvmferentia vnitaria est, qvadratvm distantiae eivs a pvncto $A$ est qvod infra scriptvm est:

(x - 1)^2 + y^2 = x^2 + y^2 - 2x + 1 = 2 - 2x

Vnde dicere abscissam per semicircvlvm svperiorem stricte decrescere idem valet ac dicere chordam qvae $A$ cvm pvncto attacto conivngit cvm arcv stricte crescere, donec post dimidiam conversionem diametrvs fiat. Cvm in lectione de rationibvs trigonometricis cosinvs ipsivs $s$ definietvr vt abscissa ipsivs $P(s)$ , haec proprietas dicet cosinvm in $\icc{0}{\pi}$ stricte decrescere: haec monotonia non dedvcitvr, sed est ipsa clavsvla dimidiae conversionis in coordinatis lecta. Iam vero ex hoc axiomate, ex factis de longitvdinibvs et areis et ex facto de reflexionibvs infra admisso dedvcitvr omne reliqvvm qvod de percvrsv adhibetvr: in hac lectione, qvomodo altera dimidia conversio se habeat et vnam conversionem non bis per idem pvnctvm transire; in lectione de fvnctionibvs trigonometricis inversis, monotonia sinvs et tangentis.

Factvm admissvm (reflexiones et conversio dimidia). Reflexio in recta distantias conservat et, cvm eis, longitvdines arcvvm, sed sensvm conversionis invertit: percvrsvm contra motvm horologii circvmferentiae cvivs centrvm est pvnctvm illivs rectae in percvrsvm secvndvm motvm horologii eivsdem longitvdinis transformat, et vice versa. Conversio dimidia circa pvnctvm, qvae est compositio reflexionvm in dvabvs rectis perpendicvlaribvs per illvd transevntibvs, distantias, longitvdines et etiam sensvm conversionis conservat.

Hoc factvm ad geometriam elementarem motvvm plani pertinet, qvam hic cvrsvs non explicat. In coordinatis reflexio in axe $x$ est transformatio $(x, y) \mapsto (x, -y)$ ; circvmferentiam vnitariam in se ipsam mittit, qvia $x^2 + (-y)^2 = x^2 + y^2$ , et pvnctvm $A = (1, 0)$ fixvm relinqvit. Ea adhibita, altera dimidia conversio percvrsvs ex prima obtinetvr.

PropositioAltera dimidia conversio

Pro omni nvmero reali $u$ pvnctvm $P(-u)$ est imago ipsivs $P(u)$ per reflexionem in axe $x$ . Praeterea, si $\pi \leq s < t \leq 2\pi$ , pvncta $P(s)$ et $P(t)$ ordinatam non positivam habent et abscissa ipsivs $P(s)$ minor est qvam ipsivs $P(t)$ : in altera sva dimidia conversione percvrsvs contra motvm horologii per semicircvlvm inferiorem transit, a sinistra ad dextram.

Percvrsvs in axe $x$ reflectitvr, et periodicitas primam dimidiam conversionem, retrorsvm percvrsam, in alteram transfert. Scribitvr $(x_w, y_w)$ pro coordinatis ipsivs $P(w)$ , et $\sigma$ pro reflexione in axe $x$ .

Demonstratio

$\sigma(x, y) = (x, \resaltar{-y}), \quad \sigma(A) = A$
Reflexio $\sigma$ signvm ordinatae mvtat, circvmferentiam vnitariam in se ipsam mittit et pvnctvm $A$ fixvm relinqvit.
$\dato{r}{P(-u) = \resaltar{\sigma(P(u))} = \left( x_u, -y_u \right)}$
Sit $u \geq 0$ . Ex facto admisso $\sigma$ percvrsvm contra motvm horologii longitvdinis $u$ , qvi a $A$ proficiscitvr et in $P(u)$ desinit, in percvrsvm secvndvm motvm horologii eivsdem longitvdinis transformat, qvi a $A$ proficiscitvr; ex vnicitate pvncti in axiomate hic percvrsvs in $P(-u)$ desinit. Si $u < 0$ , qvod modo dictvm est ad $-u$ adhibetvr et itervm reflectitvr, qvia $\sigma$ bis adhibita omne pvnctvm in svo loco relinqvit.
$P(s) = P(\resaltar{-u}) = \dato{st}{\resaltar{\left( x_u, -y_u \right)}}, \quad P(t) = \left( x_v, -y_v \right)$
Sint $\pi \leq s < t \leq 2\pi$ , et sint $u = 2\pi - s$ et $v = 2\pi - t$ , qvae $0 \leq v < u \leq \pi$ satisfacivnt. Ex periodicitate $P(s) = P(s - 2\pi) = P(-u)$ , qvod est imago ipsivs $P(u)$ ; eodem modo $P(t)$ est imago ipsivs $P(v)$ .
$x_s = \resaltar{x_u < x_v} = x_t, \quad y_s = -y_u \leq 0, \quad y_t = -y_v \leq 0$
Ex proprietate dimidiae conversionis, ad $v < u$ in $\icc{0}{\pi}$ adhibita, ordinatae ipsorvm $P(u)$ et $P(v)$ non svnt negativae et abscissa ipsivs $P(v)$ maior est qvam ipsivs $P(u)$ . Reflexio abscissas conservat et signvm ordinatarvm mvtat: qvod envntiatvm est obtinetvr.

Sit angvlvs orientatvs cvivs vertex est centrvm $O$ circvmferentiae vnitariae, et sit $A$ pvnctvm in qvo latvs initiale illam circvmferentiam secat. Cvm latvs convertitvr, pvnctvm $A$ arcvm circvmferentiae percvrrit, vt in axiomate svpra envntiato. Mensvra in radianis angvli est longitvdo illivs arcvs, signo positivo si conversio fit contra motvm horologii, negativo si secvndvm motvm horologii; si conversio plvres orbes conficit, vnaqvaeqve integra conversio longitvdinem $2\pi$ circvmferentiae addit.

ExemplvmQvidam angvli in radianis mensi

Definitio statim metiri sinit angvlos qvi simplices fractiones conversionis svnt.

Demonstratio

$L = 2\pi$
Circvmferentia vnitaria centrvm habet in vertice $O$ et radivm $1$ ; ex definitione ipsivs $\pi$ longitvdo eivs est $2\pi$ . Latvs initiale eam in pvncto $A$ secat.
$\theta = s$
Latere contra motvm horologii converso, pvnctvm $A$ arcvm percvrrit; longitvdo $s$ illivs arcvs est mensvra angvli.
$s = 1 \rightarrow \theta = 1$
Vnvs radianvs est angvlvs cvivs arcvs tantvs est qvantvs radivs. Cvm $2\pi$ pavlo maior sit qvam $6$ , vnvs radianvs pavlo minor est qvam sexta portio conversionis.
$\theta = \frac{\resaltar{2\pi}}{4} = \resaltar{\frac{\pi}{2}}$
Qvarta conversionis portio qvartam portionem circvmferentiae percvrrit: angvlvs rectvs $\frac{\pi}{2}$ radianos metitvr.
$\theta = \pi, \quad \theta = 2\pi$
Dimidia conversio dimidivm circvmferentiae percvrrit, cvivs longitvdo ex definitione est $\pi$ ; integra conversio $2\pi$ metitvr.
$\theta = -\frac{\pi}{2}$
Si latvs qvarta conversionis portione secvndvm motvm horologii convertitvr, arcvs percvrsvs idem metitvr, sed angvlvs negativvs est.

Angvlvs in radianis mensvs est nvmervs realis sine vlla vnitate mensvrae: est qvotiens longitvdinis arcvs et longitvdinis radii, qvae $1$ valet. Qvare simpliciter scribitvr $\theta = 2$ pro angvlo dvorvm radianorvm; cvm vnitas mensvrae exprimenda est, scribitvr 2 rad. Contra, mensvra in gradibvs semper svo signo indicatvr, vt in $30^\circ$ . Vice versa, omnis nvmervs realis est mensvra alicvivs angvli orientati: ex axiomate percvrsvs, pro qvoqve $\theta$ pvnctvm $A$ circvmferentiam percvrrere potest per longitvdinem $\abs{\theta}$ eo sensv qvem signvm ipsivs $\theta$ indicat, tot integros orbes conficiendo qvot opvs sit, et semirecta qvae a vertice ad pvnctvm attactvm dvcitvr est latvs terminale.

Conversio inter gradvs et radianos

Cvm integra conversio $360^\circ$ et etiam $2\pi$ radianos metiatvr, et cvm ambae mensvrae longitvdini arcvs qvem angvlvs in circvmferentia vnitaria svbtendit proportionales sint (mensvra in gradibvs ex definitione gradvs, mensvra in radianis qvia ea ipsa longitvdo est), dvae mensvrae angvli proportionales svnt.

TheoremaConversio inter gradvs et radianos

Si angvlvs $g$ gradvs et $\theta$ radianos metitvr, tvm $\theta = \frac{\pi g}{180}$ et $g = \frac{180 \theta}{\pi}$ .

Regvla ex proportione inter angvlvm et integram conversionem obtinetvr, et deinde ad notas insignes circvmferentiae adhibetvr, angvlos qvi in figvris elementaribvs apparent.

Demonstratio

$\dato{v}{360^\circ = 2\pi}$
Integra conversio $360$ gradvs et $2\pi$ radianos metitvr.
$\frac{\theta}{2\pi} = \frac{g}{360} \rightarrow \dato{c}{\theta = \resaltar{\frac{2\pi g}{360}} = \resaltar{\frac{\pi g}{180}}}$
Qvotiens angvli et integrae conversionis idem est in vtraqve mensvra, qvia ambae longitvdini arcvs proportionales svnt; vnaqvaeqve conversio addita $360^\circ$ in altera et $2\pi$ in altera addit. Solvendo pro $\theta$ regvla obtinetvr.
$1^\circ = \frac{\pi}{180}, \quad 1 = \left( \frac{180}{\pi} \right)^\circ$
Cvm $g = 1$ et cvm $\theta = 1$ : vnvs gradvs parva fractio radiani est, et vnvs radianvs pavlo plvs qvam $57$ gradvs metitvr, qvia $\frac{180}{\pi}$ est circiter 57,3.
$30^\circ = \resaltar{\frac{30\pi}{180}} = \resaltar{\frac{\pi}{6}}, \quad 45^\circ = \frac{\pi}{4}, \quad 60^\circ = \frac{\pi}{3}$
Angvli $30$ , $45$ et $60$ gradvvm, qvi in triangvlo aeqvilatero et in qvadrato apparent, fractione simplificata convertvntvr.
$90^\circ = \frac{\pi}{2}, \quad 180^\circ = \pi, \quad 270^\circ = \frac{3\pi}{2}$
Angvlvs rectvs, angvlvs planvs et tres qvartae conversionis semiaxes notant.
$120^\circ = \frac{2\pi}{3}, \quad 135^\circ = \frac{3\pi}{4}, \quad 150^\circ = \resaltar{\pi - \frac{\pi}{6}} = \resaltar{\frac{5\pi}{6}}$
Reliqvi angvli insignes conversionis addendo vel svbtrahendo obtinentvr; exempli gratia, $150^\circ$ est $180^\circ$ minvs $30^\circ$ .

In vsv, vt a gradibvs ad radianos transeatvr, per $\frac{\pi}{180}$ mvltiplicatvr, et vt a radianis ad gradvs, per $\frac{180}{\pi}$ . Sic $225^\circ = \frac{5\pi}{4}$ , $-60^\circ = -\frac{\pi}{3}$ et $\frac{7\pi}{6} = 210^\circ$ ; angvlvs $2$ radianorvm $\frac{360}{\pi}$ gradvs metitvr, circiter 114,6 gradvs.

Longitvdo arcvs et area sectoris

In circvmferentia radii $r$ angvlvs ad centrvm $\theta$ arcvm svbtendit, et dvo latera angvli, vna cvm illo arcv, sectorem circvlarem terminant. Mensvra in radianis formvlis eorvm formam praecipve simplicem dat.

TheoremaLongitvdo arcvs et area sectoris

In circvmferentia radii $r$ angvlvs ad centrvm $\theta$ radianorvm, cvm $0 < \theta \leq 2\pi$ , arcvm longitvdinis $s = r\theta$ svbtendit, et sector qvem determinat aream $A = \frac{1}{2} r^2 \theta = \frac{1}{2} rs$ habet.

Longitvdo arcvs ex proportione admissa seqvitvr. Pro area formvla primvm demonstratvr cvm angvlvs est fractio rationalis conversionis, deinde ad omnem angvlvm extenditvr per densitatem nvmerorvm rationalivm, qvae in lectione cvrsvs I de recta reali demonstrata est.

Demonstratio

$s_1 = \theta$
In circvmferentia vnitaria eodem centro angvlvs arcvm longitvdinis $\theta$ svbtendit, ex definitione radiani.
$s = r \resaltar{s_1} = \dato{s}{\resaltar{r\theta}}$
Ex proportione arcvvm ad radios arcvs respondens in circvmferentia radii $r$ est $r$ vicibvs maior.
$A = \resaltar{\frac{p}{q} \pi r^2} = \dato{f}{\resaltar{\frac{1}{2} r^2 \theta}}$
Si $\theta = \frac{2\pi p}{q}$ , cvm $p$ et $q$ natvralibvs et $p \leq q$ , discvs in $q$ sectores congrventes angvli $\frac{2\pi}{q}$ dividitvr, qvorvm vnvsqvisqve aream $\frac{\pi r^2}{q}$ habet, et sector angvli $\theta$ est vnio $p$ eorvm.
$\frac{2A}{r^2} < \varphi < \theta \rightarrow A < \resaltar{\frac{1}{2} r^2 \varphi} = A_\varphi \leq A$
Sit nvnc $\theta$ qvilibet, et ponamvs aream $A$ sectoris eivs minorem esse qvam $\frac{1}{2} r^2 \theta$ . Ex densitate nvmerorvm rationalivm exstat angvlvs $\varphi$ , fractio rationalis conversionis, inter $\frac{2A}{r^2}$ et $\theta$ . Sector eivs in sectore ipsivs $\theta$ continetvr, eivsqve area formvlam gradvs svperioris habet: ad contradictionem pervenitvr.
$A = \frac{1}{2} r^2 \theta = \frac{1}{2} r \resaltar{s}$
Eodem modo, cvm aliqvo $\varphi$ inter $\theta$ et $\frac{2A}{r^2}$ , qvi conversionem non excedit qvia sector discvm non svperat, exclvditvr $A$ maiorem esse qvam $\frac{1}{2} r^2 \theta$ . Ergo area est ea qvae indicata est, et cvm $r\theta = s$ formvla obtinetvr similis formvlae areae triangvli basis $s$ et altitvdinis $r$ .

Si angvlvs in gradibvs metiretvr, formvlae factorem conversionis secvm ferrent: arcvs esset $\frac{\pi r g}{180}$ et area sectoris $\frac{\pi r^2 g}{360}$ . Simplicitas ipsivs $s = r\theta$ praecipva cavsa est cvr radianvs praeferatvr. Praeterea formvla mensvram angvli in qvalibet circvmferentia dat, non solvm in vnitaria: $\theta = \frac{s}{r}$ est qvotiens arcvs et radii, neqve a radio electo pendet.

Angvlvs inscriptvs

Angvlvs cvivs vertex in centro circvmferentiae est angvlvs ad centrvm appellatvr. Si $A$ , $B$ et $C$ tria pvncta distincta circvmferentiae svnt, angvlvs qvem in $C$ chordae $CA$ et $CB$ efficivnt angvlvs inscriptvs appellatvr, et dicitvr arcvm extremorvm $A$ et $B$ qvi $C$ non continet svbtendere. Ex theoremate svpra demonstrato arcvi longitvdinis $s$ circvmferentiae radii $r$ respondet angvlvs ad centrvm $\frac{s}{r}$ radianorvm, etiamsi ille nvmervs $\pi$ excedat. Vt angvlvs inscriptvs cvm angvlo ad centrvm comparetvr, admittvntvr facta geometriae elementaris qvae seqvvntvr, qvae hic cvrsvs non demonstrat qvia geometriam syntheticam non explicat. Dvo prima angvlos triangvli metivntvr; dvo postrema svnt facta positionis, qvae dicvnt qvomodo angvli et arcvs distribvantvr cvm semirecta per interiorem angvli regionem transit.

Factvm admissvm (angvli triangvli et positiones in circvmferentia). Admittvntvr qvattvor affirmationes qvae infra envmerantvr, in qvibvs angvli non orientati in radianis metivntvr, inter $0$ et $\pi$ .

Svmma angvlorvm: tres angvli triangvli in svmmam angvlvm planvm efficivnt, id est $\pi$ radianos.
Triangvlvm isosceles: in triangvlo cvm dvobvs lateribvs aeqvalibvs angvli his lateribvs oppositi aeqvales svnt.
Svmma angvlorvm adiacentivm: si semirecta a vertice angvli profecta per interiorem eivs regionem transit, evm in dvos angvlos dividit qvorvm mensvrae in svmmam mensvram angvli dati efficivnt.
Positio centri: sint $A$ , $B$ et $C$ tria pvncta distincta circvmferentiae centri $O$ , et sit $D$ altervm extremvm diametri a $C$ profectae. Si $O$ neqve in chorda $CA$ neqve in chorda $CB$ est, altervm ex dvobvs accidit. Avt $O$ est intra angvlvm qvem in $C$ illae chordae efficivnt, et tvm semirecta qvae a $C$ ad $D$ dvcitvr per illam regionem interiorem transit et $D$ est in arcv extremorvm $A$ et $B$ qvi $C$ non continet. Avt $O$ est extra illvm angvlvm, et tvm altervm ex pvnctis $A$ et $B$ , qvod $A$ appellari potest, est in arcv extremorvm $D$ et $B$ qvi $C$ non continet, et semirecta qvae a $C$ ad $A$ dvcitvr per interiorem regionem angvli transit qvem in $C$ chordae $CD$ et $CB$ efficivnt.

TheoremaAngvlvs inscriptvs dimidivm arcvs svbtensi metitvr

Sint $A$ , $B$ et $C$ tria pvncta distincta circvmferentiae centri $O$ et radii $r$ , et sit $s$ longitvdo arcvs extremorvm $A$ et $B$ qvi $C$ non continet. Angvlvs inscriptvs $\gamma$ qvem in $C$ chordae $CA$ et $CB$ efficivnt est dimidivm angvli ad centrvm illi arcvi respondentis.

\gamma = \frac{s}{2r}

Demonstratio primvm tractat casvm in qvo latvs angvli inscripti per centrvm transit, et ad evm reliqvos dvos casvs facti admissi de positione centri redigit per diametrvm a $C$ profectam.

Demonstratio

$\dato{i}{\alpha = \gamma}$
Ponamvs primvm centrvm $O$ in chorda $CB$ esse, qvae tvm diametrvs est (casvs in qvo in chorda $CA$ est idem est). Sit $\alpha$ angvlvs ad $A$ triangvli verticvm $O$ , $C$ et $A$ , qvod vervm triangvlvm est qvia $A$ , a $C$ et a $B$ distinctvm, non est in recta qvae ea conivngit; hoc triangvlvm isosceles est, qvia $OA$ et $OC$ radii svnt, eivsqve angvli ad $C$ et ad $A$ aeqvales svnt.
$\delta = \pi - \left( \pi - \alpha - \gamma \right) = \resaltar{\alpha + \gamma} = \dato{d}{\resaltar{2\gamma}} < \pi$
Sit $\delta$ angvlvs ad centrvm qvem radii $OA$ et $OB$ efficivnt. Cvm $C$ , $O$ et $B$ in vna recta iaceant, $\delta$ et angvlvs triangvli ad $O$ in svmmam $\pi$ efficivnt; cvm tres angvli triangvli etiam in svmmam $\pi$ efficiant, $\delta$ est svmma reliqvorvm dvorvm. Praeterea angvlvs ad $O$ positivvs est, ita vt $\alpha + \gamma$ minor sit qvam $\pi$ , et item $\delta$ .
$s = r \resaltar{\delta} = \resaltar{2r\gamma} \rightarrow \dato{c}{\gamma = \frac{s}{2r}}$
Radii $OA$ et $OB$ circvmferentiam in dvos arcvs dividvnt; cvm $\delta < \pi$ , ex theoremate arcvs, is qvem $\delta$ svbtendit $r\delta$ metitvr, minvs qvam semicircvlvs, alter plvs. Pvnctvm $C$ , extremvm diametri ipsi $B$ oppositvm, semicircvlo a $B$ distat in vtroqve sensv, ita vt in arcv minore non sit: hic est arcvs qvem $\gamma$ svbtendit, et $s = r\delta$ .
$\gamma = \frac{s_1}{2r} + \frac{s_2}{2r} = \frac{\resaltar{s_1 + s_2}}{2r} = \frac{\resaltar{s}}{2r}$
Si centrvm est intra angvlvm, sit $D$ altervm extremvm diametri a $C$ profectae. Ex facto admisso de positione centri semirecta a $C$ ad $D$ angvlvm in dvos dividit, qvorvm vterqve latvs per centrvm transiens habet, qvorvmqve mensvrae in svmmam $\gamma$ efficivnt ex svmma angvlorvm adiacentivm; et $D$ arcvm svbtensvm in dvos arcvs dividit, longitvdinvm $s_1$ et $s_2$ , qvi in svmmam $s$ efficivnt. Ex casv iam tractato, ad vtrvmqve angvlvm adhibito, obtinetvr qvod infra scriptvm est.
$\gamma = \frac{s_2}{2r} - \frac{s_1}{2r} = \frac{\resaltar{s_2 - s_1}}{2r} = \frac{\resaltar{s}}{2r}$
Si centrvm est extra angvlvm, sit rvrsvs $D$ altervm extremvm diametri a $C$ profectae. Ex facto admisso de positione centri poni potest $A$ esse in arcv qvi a $D$ ad $B$ dvcitvr non transiens per $C$ , et semirectam a $C$ ad $A$ angvlvm latervm $CD$ et $CB$ dividere in angvlvm latervm $CD$ et $CA$ et in $\gamma$ . Ille arcvs, longitvdinis $s_2$ , constat ex arcv qvi a $D$ ad $A$ dvcitvr, longitvdinis $s_1$ , et ex arcv qvem $\gamma$ svbtendit; et $\gamma$ est differentia dvorvm angvlorvm qvorvm vterqve latvs per centrvm transiens habet.

CorollarivmAngvli inscripti in eadem chorda

Dvo angvli inscripti iisdem extremis $A$ et $B$ aeqvales svnt si vertices eorvm in eodem ex dvobvs arcvbvs svnt qvos chorda $AB$ determinat, svpplementarii si in arcvbvs diversis. Praesertim omnis angvlvs semicircvlo inscriptvs, id est cvivs extrema $A$ et $B$ svnt extrema diametri, rectvs est.

Satis est in vnoqvoqve casv legere qvem arcvm vnvsqvisqve angvlvs svbtendat, et theorema adhibere. Sint $C$ et $C'$ vertices, $\gamma$ et $\gamma'$ angvli, et $s$ longitvdo arcvs extremorvm $A$ et $B$ qvi $C$ non continet.

Demonstratio

$\dato{g}{\gamma = \frac{s}{2r}} = \gamma'$
Si $C'$ in eodem arcv est ac $C$ , ambo angvli altervm arcvm svbtendvnt, longitvdinis $s$ , et ex theoremate idem metivntvr.
$\gamma + \gamma' = \frac{s}{2r} + \frac{\resaltar{2\pi r - s}}{2r} = \resaltar{\pi}$
Si $C'$ in altero arcv est, arcvm svbtendit qvi $C$ continet, cvivs longitvdo est id qvod ipsi $s$ deest ad circvmferentiam complendam; dvae mensvrae in svmmam angvlvm planvm efficivnt.
$s = \pi r \rightarrow \gamma = \frac{\resaltar{\pi r}}{2r} = \resaltar{\frac{\pi}{2}}$
Si $AB$ diametrvs est, ambo arcvs semicircvli svnt, longitvdinis $\pi r$ , et angvlvs rectvs est; hic casvs particvlaris theorema Thaletis appellatvr.

Pvncta qvae in circvmferentia non svnt chordam svb aliis angvlis vident. Effectvs qvi seqvitvr etiam ex geometria elementari admittitvr: obtinetvr ex theoremate svpra demonstrato et ex propositione 21 libri I Elementorvm Evclidis, ex qva pvnctvm intra triangvlvm vnvm ex lateribvs eivs svb angvlo maiore videt qvam vertex oppositvs; sed demonstratio eivs postvlat vt sciatvr vbi segmenta a pvncto profecta circvmferentiam secent, qvaestio positionis qvam hic cvrsvs non explicat. Adhibebitvr in postremo problemate solvto lectionis de fvnctionibvs trigonometricis inversis.

Factvm admissvm (angvli a pvnctis interioribvs et exterioribvs). Sint $A$ et $B$ dvo pvncta circvmferentiae, et $\gamma$ mensvra commvnis angvlorvm inscriptorvm extremis $A$ et $B$ qvorvm vertex in altero ex dvobvs arcvbvs est qvos haec pvncta determinant. Si $Q$ est pvnctvm qvod in circvmferentia non est, ex eodem latere rectae $AB$ sitvm ac ille arcvs, angvlvs qvem in $Q$ segmenta $QA$ et $QB$ efficivnt minor est qvam $\gamma$ si $Q$ extra discvm est, et maior qvam $\gamma$ si intra.

Positio normalis et angvli coterminales

In plano Cartesiano, in lectione cvrsvs I de aeqvatione rectae et systematibvs Cartesianis investigato, vnvsqvisqve angvlvs positionem praecipvam habet.

Angvlvs orientatvs est in positione normali si vertex eivs est origo et latvs initiale eivs est semiaxis $x$ positivvs. Latvs terminale eivs tvm est semirecta qvae ab origine proficiscitvr et per pvnctvm $P(\theta)$ axiomatis percvrsvs transit, id est pvnctvm circvmferentiae vnitariae $x^2 + y^2 = 1$ ad qvod pervenitvr percvrrendo, a $(1, 0)$ , arcvm longitvdinis $\abs{\theta}$ eo sensv qvem signvm ipsivs $\theta$ indicat. Dvo angvli svnt coterminales si, in positione normali, idem latvs terminale habent.

Axes planvm in qvattvor qvadrantes dividvnt, contra motvm horologii nvmeratos ab eo qvi pvncta coordinatarvm positivarvm continet. Angvlvs in positione normali est in qvadrante qvi latvs terminale eivs continet; si hoc in axem cadit, angvlvs qvadrantalis appellatvr. Coordinatae pvncti $P(\theta)$ svnt cosinvs et sinvs ipsivs $\theta$ , qvae in proxima lectione investigabvntvr, de rationibvs trigonometricis.

Integra conversio, in vtrvmvis sensvm, latvs ad positionem svam redvcit, ex periodicitate percvrsvs: angvlvs $\theta + 2\pi$ coterminalis est cvm $\theta$ , et item $\theta - 2\pi$ , $\theta + 4\pi$ et, generatim, $\theta + 2k\pi$ cvm $k$ integro. Vt sciatvr vbi qvilibet angvlvs desinat, satis est igitvr integras conversiones dedvcere, vt mvltipla divisoris dedvcvntvr in lectione cvrsvs I de divisibilitate et divisione cvm residvo. Theorema qvod seqvitvr est forma realis illivs divisionis: divisor est $2\pi$ , qvotiens est nvmervs conversionvm et residvvm angvlvs qvi restat.

TheoremaRedvctio ad primam conversionem

Pro omni nvmero reali $\theta$ exstant vnicvs nvmervs integer $k$ et vnicvs nvmervs realis $\theta_0$ tales vt $\theta = \theta_0 + 2k\pi$ et $0 \leq \theta_0 < 2\pi$ .

Exsistentia nititvr proprietate Archimedea, in lectione cvrsvs I de expressione decimali nvmerorvm rationalivm demonstrata, et vnicitas eo qvod nvmervs integer stricte inter $-1$ et $1$ comprehensvs nvllvs est.

Demonstratio

$\dato{k}{k \leq \frac{\theta}{2\pi} < k + 1}$
Sit $\theta$ nvmervs realis. Ex proprietate Archimedea et principio bonae ordinationis, vt in demonstratione densitatis nvmerorvm rationalivm in lectione de recta reali, exstat minimvs nvmervs integer $m$ maior qvam $\frac{\theta}{2\pi}$ ; sit $k = m - 1$ , qvi non iam maior est qvam ille nvmervs.
$\resaltar{2k\pi \leq \theta < 2k\pi + 2\pi} \rightarrow \resaltar{0 \leq \theta - 2k\pi < 2\pi}$
Per $2\pi$ , qvi positivvs est, mvltiplicamvs, et $2k\pi$ ab omnibvs tribvs membris svbtrahimvs.
$\dato{e}{\theta = \theta_0 + 2k\pi, \quad \theta_0 \in \ico{0}{2\pi}}$
Sit $\theta_0 = \theta - 2k\pi$ . Tvm $\theta$ constat ex $k$ integris conversionibvs qvas seqvitvr angvlvs $\theta_0$ , qvi in prima conversione est: $\theta$ et $\theta_0$ coterminales svnt.
$\dato{d}{2\pi(k - j) = \theta_1 - \theta_0}, \quad -2\pi < \theta_1 - \theta_0 < 2\pi$
Pro vnicitate ponamvs etiam $\theta = \theta_1 + 2j\pi$ , cvm $j$ integro et $0 \leq \theta_1 < 2\pi$ . Dvabvs scriptvris svbtractis, differentia $\theta_1 - \theta_0$ est mvltiplvm integrvm ipsivs $2\pi$ ; et cvm ambo nvmeri in intervallo $\ico{0}{2\pi}$ sint, illa differentia inter $-2\pi$ et $2\pi$ iacet.
$\abs{k - j} < 1 \rightarrow k = j \rightarrow \theta_1 = \theta_0$
Per $2\pi$ dividendo, nvmervs integer $k - j$ stricte inter $-1$ et $1$ iacet: nvllvs est. Ergo $j = k$ et, ex aeqvalitate svpra obtenta, $\theta_1 = \theta_0$ .
$\theta_0 = \resaltar{-\frac{23\pi}{6} + 4\pi} = \resaltar{\frac{\pi}{6}}$
Exempli gratia, pro $\theta = -\frac{23\pi}{6}$ habetvr $\frac{\theta}{2\pi} = -\frac{23}{12}$ , qvi inter $-2$ et $-1$ iacet: est $k = -2$ , et angvlvs fere dvos orbes secvndvm motvm horologii conficit vt in latere ipsivs $\frac{\pi}{6}$ desinat.

Nvmervs $\theta_0$ redvctio ipsivs $\theta$ ad primam conversionem appellatvr. In gradibvs eadem ratiocinatio, cvm $360$ loco ipsivs $2\pi$ , qvemlibet angvlvm ad angvlvm inter $0^\circ$ et $360^\circ$ redigit, hoc exclvso: exempli gratia, $1000^\circ = 280^\circ + 2 \cdot 360^\circ$ . Redvctione agnoscvntvr omnes angvli qvi in eodem latere desinvnt; antea sciendvm est vnam conversionem non bis per idem pvnctvm transire, qvod ex axiomate percvrsvs dedvcitvr.

LemmaVna conversio pvncta non iterat

Si $0 \leq \alpha < \beta < 2\pi$ , tvm $P(\alpha) \neq P(\beta)$ .

Casvs separantvr secvndvm dimidiam conversionem in qva $\alpha$ et $\beta$ svnt: in vtraqve abscissa stricte monotona est; ab altera ad alteram signvm ordinatae mvtatvr. Scribitvr $(x_s, y_s)$ pro coordinatis ipsivs $P(s)$ .

Demonstratio

$0 < s < \pi \vee \pi < s < 2\pi \rightarrow \dato{x}{-1 < x_s < 1}$
Ex proprietate dimidiae conversionis, cvm $s$ a $0$ ad $\pi$ crescit, abscissa ipsivs $P(s)$ a $1$ ad $-1$ stricte decrescit; ex propositione de altera dimidia conversione, cvm $s$ a $\pi$ ad $2\pi$ crescit, ea a $-1$ stricte crescit vsqve ad abscissam ipsivs $P(2\pi)$ , qvod ex periodicitate est $P(0) = (1, 0)$ .
$y_s^2 = \resaltar{1 - x_s^2} > 0 \rightarrow \dato{y}{\pi < s < 2\pi \rightarrow y_s < 0}$
In his casibvs ordinata nvlla non est, qvia qvadratvm eivs est $1 - x_s^2 > 0$ : positiva est in prima dimidia conversione, qvia negativa non est, et negativa in altera, qvia positiva non est.
$x_\alpha \neq x_\beta \rightarrow P(\alpha) \neq P(\beta)$
Si $\alpha$ et $\beta$ ambo in $\icc{0}{\pi}$ svnt, vel ambo in $\ico{\pi}{2\pi}$ , dvo pvncta abscissas diversas habent, ex stricta monotonia in vtraqve dimidia conversione.
$\resaltar{y_\beta < 0 \leq y_\alpha} \rightarrow P(\alpha) \neq P(\beta)$
Si $\alpha < \pi < \beta$ , pvnctvm $P(\beta)$ ordinatam negativam habet, cvm ordinata ipsivs $P(\alpha)$ , in prima dimidia conversione, negativa non sit. In omnibvs casibvs dvo pvncta distincta svnt.

LemmaAngvli coterminales

Dvo angvli $\theta$ et $\varphi$ coterminales svnt si et solvm si differentia eorvm est mvltiplvm integrvm ipsivs $2\pi$ , id est si exstat nvmervs integer $k$ talis vt $\theta - \varphi = 2k\pi$ . Praesertim coterminales svnt si et solvm si eandem redvctionem ad primam conversionem habent.

Vnvsqvisqve angvlvs sva redvctione svbstitvitvr, et lemma modo demonstratvm redvctiones comparat.

Demonstratio

$\dato{p}{P(\theta) = P(\resaltar{\theta_0}), \quad P(\varphi) = P(\resaltar{\varphi_0})}$
Ex theoremate redvctionis $\theta = \theta_0 + 2k\pi$ et $\varphi = \varphi_0 + 2j\pi$ , cvm $\theta_0$ et $\varphi_0$ in $\ico{0}{2\pi}$ ; ex periodicitate percvrsvs integrae conversiones pvnctvm non mvtant, et latvs terminale cvivsqve angvli est latvs redvctionis eivs.
$\dato{c}{P(\theta) = P(\varphi) \leftrightarrow \theta_0 = \varphi_0}$
Si $\theta_0 \neq \varphi_0$ , ex lemmate modo demonstrato pvncta $P(\theta_0)$ et $P(\varphi_0)$ distincta svnt, et item latera terminalia; si aeqvales svnt, latera congrvvnt. Ergo angvli coterminales svnt si et solvm si eandem redvctionem habent.
$\theta_0 = \varphi_0 \rightarrow \theta - \varphi = \resaltar{2(k - j)\pi}, \qquad \theta - \varphi = 2m\pi \rightarrow \theta = \varphi_0 + \resaltar{2(j + m)\pi} \rightarrow \theta_0 = \varphi_0$
Si redvctiones aeqvales svnt, differentia est mvltiplvm integrvm ipsivs $2\pi$ . Vice versa, si $\theta - \varphi = 2m\pi$ , tvm $\theta = \varphi_0 + 2(j + m)\pi$ , et ex vnicitate redvctionis $\theta_0 = \varphi_0$ .

Ex his seqvitvr angvlos coterminales cvm $\theta$ esse prorsvs eos formae $\theta + 2k\pi$ , cvm $k$ integro, et qvadrantem angvli ex redvctione eivs decerni: est in primo qvadrante si $0 < \theta_0 < \frac{\pi}{2}$ , in secvndo si $\frac{\pi}{2} < \theta_0 < \pi$ , in tertio si $\pi < \theta_0 < \frac{3\pi}{2}$ et in qvarto si $\frac{3\pi}{2} < \theta_0 < 2\pi$ ; qvadrantalis est si $\theta_0$ est $0$ , $\frac{\pi}{2}$ , $\pi$ avt $\frac{3\pi}{2}$ . Sic $-\frac{23\pi}{6}$ in primo qvadrante est, et $1000^\circ$ , cvivs redvctio est $280^\circ$ , in qvarto.

Postremo omne pvnctvm circvmferentiae vnitariae est pvnctvm $P(\theta)$ alicvivs angvli: haec est proprietas operimenti axiomatis percvrsvs, qvae dicit circvmferentiam a $(1, 0)$ percvrrendo per omnia eivs pvncta transiri. Qvod si ita est, ex redvctione ad primam conversionem, ex periodicitate et ex lemmate de angvlis coterminalibvs, vnvmqvodqve pvnctvm circvmferentiae vnitariae est $P(\theta)$ pro vnico $\theta$ in $\ico{0}{2\pi}$ , et angvli qvi ei respondent svnt prorsvs ii qvi cvm illo $\theta$ coterminales svnt.

Velocitas angvlaris et velocitas linearis

Cvm corpvs circa axem convertitvr, omnia eivs pvncta arcvs circvmferentiarvm circa axem describvnt, et omnia eodem tempore per evndem angvlvm convertvntvr. Si tempore $t$ corpvs, aeqvabili celeritate, per angvlvm $\theta$ convertitvr, velocitas angvlaris eivs est qvotiens $\omega = \frac{\theta}{t}$ , qvae radianis in secvndvm metitvr; exprimitvr etiam conversionibvs in minvtvm, et cvm vna conversio sit $2\pi$ radiani et vnvm minvtvm $60$ secvnda, vna conversio in minvtvm aeqvat $\frac{\pi}{30}$ radianos in secvndvm.

Pvnctvm in distantia $r$ ab axe sitvm illo tempore arcvm longitvdinis $s = r\theta$ percvrrit, ex theoremate longitvdinis arcvs. Velocitas linearis eivs, qvotiens distantiae percvrsae et temporis, est igitvr prodvctvm radii et velocitatis angvlaris:

v = \frac{s}{t} = \frac{r\theta}{t} = r\omega

Ex his seqvitvr in corpore qvod convertitvr velocitatem angvlarem eandem esse pro omnibvs pvnctis, linearem vero pro distantia ab axe crescere: margo disci qvi convertitvr celerivs movetvr qvam pvncta centro propinqva. Haec relatio, qvae valet tantvm si angvli in radianis metivntvr, initivm est investigationis motvs circvlaris in cvrsv Fvndamenta Mechanicae. Dvae rotae loro vel catena qvae non labitvr conivnctae, vel qvae inter se dentibvs implicantvr, in pvncto contactvs eandem velocitatem linearem habent; ergo $\omega_1 r_1 = \omega_2 r_2$ , et rota minor celerivs convertitvr, vt in problemate solvto 3 videbitvr.

Problemata solvta

Problemata Terram ex angvlo metivntvr, aream segmenti circvli compvtant, conversionem per catenam transmittvnt, sectorem maximae areae perimetro dato qvaervnt et nvmerant qvotiens indices horologii inter se congrvant. Vnvmqvodqve propriam notionem postvlat, qvam qvaerere expedit anteqvam solvtio legatvr.

Problema solvtvm 1Mensvra Terrae secvndvm Eratosthenem

Circa annvm 240 ante aeram vvlgarem Eratosthenes animadvertit meridie solstitii aestivi Solem fvndvm pvteorvm Syenes illvstrare, cvm Alexandriae, qvae ad septentriones sita est et, vt ipse svpposvit, in eodem meridiano, radii cvm linea verticali angvlvm 7 gradvvm et 12 minvtorvm efficerent. Svmptis 800 km vt distantia inter dvas vrbes, compvtare radivm Terrae et longitvdinem meridiani, longitvdinem arcvs meridiani qvi vni minvto respondet et distantiam, per meridianvm, ab aeqvatore ad pvnctvm latitvdinis $33^\circ$ .

Solvtio

$\alpha = 7^\circ + \resaltar{\left( \frac{12}{60} \right)^\circ} = \resaltar{\left( \frac{36}{5} \right)^\circ}$
Angvlvs, in gradibvs expressvs, est $7$ plvs dvodecim sexagesimae; est qvinqvagesima portio conversionis, qvia $\frac{36}{5} \cdot 50 = 360$ .
$\theta = \frac{\resaltar{\frac{36}{5}} \pi}{180} = \dato{t}{\resaltar{\frac{\pi}{25}}}$
Cvm radii Solis paralleli adveniant et Syenae in directione radii terrestris cadant, angvlvs Alexandriae inter radivm lvminis et lineam verticalem aeqvalis est angvlo qvem in centro Terrae radii terrestres dvarvm vrbivm efficivnt: svnt angvli alterni inter parallelas. In radianis, ex regvla conversionis, est qvod infra scriptvm est.
$R = \frac{s}{\theta} = \frac{\resaltar{800}}{\resaltar{\frac{\pi}{25}}} = \dato{r}{\resaltar{\frac{20000}{\pi}}}$
Illa 800 km svnt arcvs meridiani qvem ille angvlvs ad centrvm svbtendit. Ex $s = R\theta$ radivs solvitvr, circiter 6366 km.
$2\pi R = 2\pi \cdot \resaltar{\frac{20000}{\pi}} = \resaltar{40000}$
Longitvdo meridiani est longitvdo circvmferentiae radii $R$ ; est etiam $50$ vicibvs distantia inter dvas vrbes.
$s = \frac{20000}{\pi} \cdot \resaltar{\frac{\pi}{10800}} = \resaltar{\frac{50}{27}}$
Minvtvm est sexagesima portio gradvs, id est $\frac{\pi}{10800}$ radiani. Arcvs eivs circiter 1,852 km metitvr: est milliarivm navticvm, qvod prorsvs hoc modo definitvm est.
$s = \frac{20000}{\pi} \cdot \resaltar{\frac{11\pi}{60}} = \resaltar{\frac{11000}{3}}$
Latitvdo pvncti est angvlvs qvem radivs eivs cvm plano aeqvatoris efficit; $33^\circ$ svnt $\frac{11\pi}{60}$ radiani. Distantia qvaesita est circiter 3667 km.

Eratosthenes distantiam stadiis expressit, qvorvm valor accvratvs dispvtatvr; nvmero hic adhibito meridianvs 40000 km metitvr, neqve casv: exevnte saecvlo XVIII metrvm definitvm est vt decies millies millesima portio qvadrantis meridiani. Cetervm Alexandria non prorsvs ad septentriones Syenes iacet, sed circiter tribvs gradibvs magis ad occidentem, ita vt arcvs qvi eas conivngit non omnino arcvs meridiani sit.

Problema solvtvm 2Segmentvm circvli sexaginta gradvvm

Chorda extremorvm $A$ et $B$ , in circvmferentia centri $O$ et radii $r$ , angvlvm ad centrvm $\frac{\pi}{3}$ svbtendit. Compvtare aream segmenti circvli qvod chorda et arcv minore extremorvm $A$ et $B$ terminatvr, et fractionem disci qvam occvpat.

Solvtio

$A_s = \frac{1}{2} r^2 \cdot \resaltar{\frac{\pi}{3}} = \dato{s}{\resaltar{\frac{\pi r^2}{6}}}$
Segmentvm circvli est id qvod ex sectore verticis $O$ et extremorvm $A$ et $B$ restat triangvlo verticvm $O$ , $A$ et $B$ dempto. Aream sectoris dat theorema sectoris.
$\frac{\resaltar{\pi - \frac{\pi}{3}}}{2} = \resaltar{\frac{\pi}{3}}$
Illvd triangvlvm dvo latera radio aeqvalia habet: isosceles est, eivsqve angvli ad $A$ et ad $B$ aeqvales svnt. Cvm tres angvli triangvli in svmmam $\pi$ efficiant, ex facto in hac lectione admisso, vnvsqvisqve eorvm $\frac{\pi}{3}$ metitvr, et triangvlvm aeqvilatervm est, lateris $r$ .
$h = \frac{\sqrt{3}}{2} r, \quad A_t = \frac{1}{2} r \cdot \resaltar{\frac{\sqrt{3}}{2} r} = \dato{t}{\resaltar{\frac{\sqrt{3}}{4} r^2}}$
Altitvdo a $O$ dvcta basim in pvncto medio secat; ex theoremate Pythagorae qvadratvm eivs est $r^2 - \frac{r^2}{4}$ . Area triangvli est dimidivm basis in altitvdinem dvctae.
$A = \resaltar{\frac{\pi r^2}{6} - \frac{\sqrt{3}}{4} r^2} = \dato{a}{\resaltar{\frac{2\pi - 3\sqrt{3}}{12} r^2}}$
Area segmenti circvli est differentia, cvm denominatore commvni $12$ ; positiva est, qvia $2\pi > 6 > 3\sqrt{3}$ .
$\frac{A}{\pi r^2} = \frac{2\pi - 3\sqrt{3}}{12\pi}$
Per aream disci dividendo, segmentvm circvli circiter 2,9 centesimas disci occvpat; sector sextam eivs portionem occvpat, et triangvlvm fere 83 centesimas sectoris implet.

Problema solvtvm 3Catena birotae

In birota catena orbem dentatvm, radii $10$ cm, qvem pedalia movent, cvm rotvla dentata conivngit, radii $4$ cm, axi rotae posterioris affixa, cvivs radivs est $35$ cm. Si homo birota vectvs singvlis secvndis vnam conversionem pedalivm efficit, compvtare velocitatem angvlarem rotae et velocitatem birotae, posito catenam non labi et rotam sine lapsv volvi.

Solvtio

$\omega_1 = \dato{w}{2\pi}$
Vna conversio in secvndvm svnt $2\pi$ radiani in secvndvm: haec est velocitas angvlaris orbis dentati.
$v = r_1 \omega_1 = 10 \cdot \resaltar{2\pi} = \dato{v}{\resaltar{20\pi}}$
Tota catena eadem celeritate movetvr, qvae est velocitas linearis marginis orbis dentati, in centimetris in secvndvm.
$\omega_2 = \frac{v}{r_2} = \frac{\resaltar{20\pi}}{\resaltar{4}} = \dato{p}{\resaltar{5\pi}}$
Catena marginem rotvlae dentatae eadem velocitate lineari trahit; ex $\omega_1 r_1 = \omega_2 r_2$ velocitas angvlaris rotvlae dentatae obtinetvr: dvae conversiones et dimidia in secvndvm.
$v_r = 35 \cdot \resaltar{5\pi} = \dato{b}{\resaltar{175\pi}}$
Rota axi rotvlae dentatae affixa est et eadem velocitate angvlari convertitvr, sed margo eivs mvlto longivs ab axe abest, eivsqve velocitas linearis mvlto maior est, in centimetris in secvndvm.
$175\pi \cdot \resaltar{\frac{3600}{100000}} = \resaltar{\frac{63\pi}{10}}$
Si rota non labitvr, in vnaqvaqve conversione birota tantvm progreditvr qvanta est circvmferentia rotae, qvae pavlatim in solo ponitvr: velocitas eivs est velocitas marginis rotae. Vnvm centimetrvm in secvndvm svnt $\frac{3600}{100000}$ chiliometra in horam, ita vt birota circiter 19,8 chiliometra in horam progrediatvr.

Problema solvtvm 4Sector perimetri fixi et areae maximae

Filo ferreo longitvdinis $P$ ambitvs sectoris circvlaris formatvr: dvo radii et arcvs qvi eos conivngit. Invenire radivm et angvlvm sectoris maximae areae, et valorem illivs areae.

Solvtio

$P = 2r + s \rightarrow \dato{s}{s = P - 2r}$
Si radivs est $r$ et angvlvs $\theta$ , ambitvs constat ex dvobvs radiis et arcv longitvdinis $s = r\theta$ .
$A = \frac{1}{2} r \resaltar{s} = \frac{1}{2} r \resaltar{(P - 2r)} = \dato{a}{\frac{Pr}{2} - r^2}$
Ex theoremate sectoris area est dimidivm radii in arcvm dvcti: est fvnctio qvadratica radii.
$\dato{d}{\frac{P}{2 + 2\pi} \leq r < \frac{P}{2}}$
Arcvs debet esse positivvs nec maior qvam tota circvmferentia, id est $0 < \theta \leq 2\pi$ ; ex $P - 2r > 0$ et $P - 2r \leq 2\pi r$ intervallvm radiorvm possibilivm oritvr.
$A = \resaltar{\frac{P^2}{16} - \left( r - \frac{P}{4} \right)^2} \leq \resaltar{\frac{P^2}{16}}$
Qvadrato completo, area est nvmervs fixvs minvs qvadratvm. Delineatio eivs, vt fvnctio ipsivs $r$ , est parabola $\left( r - \frac{P}{4} \right)^2 = -\left( A - \frac{P^2}{16} \right)$ , qvae deorsvm aperitvr; ex theoremate verticis lectionis cvrsvs I de characterizatione parabolarvm pvnctvm eivs altissimvm est vertex.
$r = \frac{P}{4}, \quad A = \frac{P^2}{16}$
Abscissa verticis, $\frac{P}{4}$ , ad intervallvm radiorvm possibilivm pertinet, qvia $2 + 2\pi > 4$ . Ergo area maxima prorsvs illo radio attingitvr.
$\theta = \frac{s}{r} = \frac{\resaltar{P - \frac{P}{2}}}{\frac{P}{4}} = \resaltar{2}$
Arcvs respondens $\frac{P}{2}$ metitvr, et angvlvs est qvotiens arcvs et radii: $2$ radiani, circiter 114,6 gradvs, qvicvmqve sit $P$ . Mirvm in modvm area maxima est area qvadrati perimetri $P$ .

Problema solvtvm 5Indices horologii

Compvtare angvlvm qvem indices horologii hora 3:40 efficivnt. Deinde demonstrare, inter horam 12:00 et horam 12:00 proximam (hac excepta), indices prorsvs vndecies inter se congrvere, intervallis aeqvalibvs $\frac{12}{11}$ horarvm separatos.

Solvtio

$\dato{p}{\alpha_m = 360t, \quad \alpha_h = 30t}$
Positionem cvivsqve indicis metimvr angvlo, in gradibvs et secvndvm motvm horologii, per qvem ab hora 12:00 conversvs est. Post $t$ horas index minvtorvm $t$ orbes confecit, et index horarvm, qvi singvlis 12 horis vnvm orbem conficit, $30^\circ$ in horam conversvs est.
$\alpha_m = 360 \cdot \frac{11}{3} = \resaltar{1320^\circ} = \resaltar{240^\circ + 3 \cdot 360^\circ}$
Hora 3:40 transiervnt $3 + \frac{40}{60} = \frac{11}{3}$ horae. Index minvtorvm tres integros orbes confecit, qvi positionem eivs non mvtant, et in redvctione angvli svi stat.
$\alpha_h = 30 \cdot \resaltar{\frac{11}{3}} = \resaltar{110^\circ}$
Index horarvm nvllvm orbem complevit.
$240^\circ - 110^\circ = \resaltar{130^\circ} = \resaltar{\frac{13\pi}{18}}$
Differentia positionvm redactarvm minor est qvam $180^\circ$ , ita vt sit angvlvs qvem indices efficivnt, in gradibvs et in radianis.
$360t - 30t = 330t = 360k \leftrightarrow \dato{t}{t = \frac{12k}{11}}, \quad k \in \mathbb{Z}$
Indices inter se congrvvnt cvm angvli eorvm coterminales svnt, id est, ex lemmate de angvlis coterminalibvs, cvm differentia eorvm est mvltiplvm integrvm vnivs conversionis.
$0 \leq \frac{12k}{11} < 12 \leftrightarrow 0 \leq k < 11$
Condicio $0 \leq t < 12$ aeqvivalet ipsi $0 \leq k < 11$ : svnt prorsvs vndecim congrventiae, pro $k = 0, 1, \ldots, 10$ , et dvae continvae $\frac{12}{11}$ horis distant, circiter vna hora, 5 minvtis et 27 secvndis. Positiones indicis horarvm, $\frac{360k}{11}$ gradvs, svnt vertices hendecagoni regvlaris.

Problemata proposita

Problemata notiones lectionis seqvvntvr: sectorem in convm vertere, longitvdinem spirae circvmferentias svmmando aestimare, taeniam in portiones rectas et arcvs dividere qvorvm angvli ad centrvm addvntvr, et angvlos ad primam conversionem per angvlos coterminales redigere vt positiones et momenta nvmerentvr. Vnvmqvodqve indicivm habet, qvod aperire expedit solvm postqvam problema temptatvm est; qvae demonstrationem postvlant solvtionem explicatam praeterea habent.

Problema propositvm 1Convs chartacevs

Ex folio sector circvlaris radii $12$ cm et angvli $120^\circ$ exciditvr, et dvo eivs radii conglvtinantvr vt svperficies lateralis coni formetvr. Compvtare radivm basis et altitvdinem coni.

Indicivm

Arcvs sectoris fit circvmferentia basis, et radivs sectoris fit distantia verticis coni a qvolibet pvncto illivs circvmferentiae. Altitvdo deinde theoremate Pythagorae obtinetvr.

Responsvm

Radivs basis est $4$ cm et altitvdo $8\sqrt{2}$ cm.

Problema propositvm 2Semita disci compacti

In disco compacto notitia per vnicam semitam spiralem inscripta est, qvae 2,3 cm a centro incipit et ad 5,8 cm desinit; dvo gyri continvi spirae 1,6 micrometra distant. Capvt legens semitam velocitate lineari constanti 1,2 metrorvm in secvndvm percvrrit. Aestimare longitvdinem semitae et dvrationem lectionis integrae, vnvmqvemqve gyrvm spirae circvmferentia circa axem disci posita approximando.

Indicivm

Si $r$ et $R$ svnt radii initialis et finalis et $p$ est intervallvm inter gyros, svnt $N = \frac{R - r}{p}$ gyri, radiorvm $r$ , $r + p$ , $r + 2p$ , et sic deinceps. Longitvdines eorvm progressionem arithmeticam efficivnt, cvivs svmma in lectione cvrsvs I de svmmationibvs, progressionibvs et indvctione compvtata est; effectvs proximvs est areae coronae per intervallvm inter gyros divisae. Attende ad vnitates mensvrarvm.

Responsvm

Spira $21875$ gyros conficit, eivsqve longitvdo est circiter qvod infra scriptvm est, circiter 5,57 km; lectio integra circiter 4640 secvnda dvrat, id est circiter 77 minvta.

L \approx \frac{\pi \left( R^2 - r^2 \right)}{p}

Problema propositvm 3Taenia qvae tres tvbos ligat

Tres tvbi cylindrici aeqvales, radii $r$ , paralleli collocantvr, vnvsqvisqve in dvobvs reliqvis innixvs, et taenia tensa qvae eos circvmdat ligantvr. Demonstrare taeniam $6r + 2\pi r$ metiri, et longitvdinem eivs compvtare pro tvbis radii 5 cm.

Indicivm

In sectione tvbis perpendicvlari vnaqvaeqve portio recta taeniae rectangvlvm efficit cvm radiis ad pvncta contactvs dvctis et cvm latere triangvli centrorvm. Pro arcvbvs compvta angvlvm ad centrvm qvem taenia in vnoqvoqve tvbo amplectitvr, et adhibe svmmam angvlorvm triangvli.

Responsvm

Taenia constat ex tribvs portionibvs rectis longitvdinis $2r$ et ex tribvs arcvbvs qvi simvl integram circvmferentiam efficivnt. Pro $r = 5$ metitvr $30 + 10\pi$ cm, circiter 61,4 cm.

Solvtio explicata

Solvtio

$\operatorname{d}(A, B) = \operatorname{d}(B, C) = \operatorname{d}(C, A) = \dato{l}{2r}$
Fascis plano tvbis perpendicvlari secatvr: videntvr tres circvmferentiae radii $r$ , centrorvm $A$ , $B$ et $C$ . Dvae circvmferentiae extrinsecvs tangentes centra habent distantia svmmae radiorvm aeqvali, ita vt triangvlvm centrorvm aeqvilatervm sit, lateris $2r$ .
$\operatorname{d}(T, T') = \resaltar{\operatorname{d}(A, B)} = \resaltar{2r} \rightarrow \dato{t}{3 \cdot 2r = 6r}$
Vnaqvaeqve portio recta taeniae dvos tvbos tangit, exempli gratia eos centrorvm $A$ et $B$ , in dvobvs pvnctis $T$ et $T'$ . Radivs ad pvnctvm contactvs dvctvs tangenti perpendicvlaris est, ex lectione cvrsvs I de circvmferentia; radii $AT$ et $BT'$ eidem rectae perpendicvlares svnt, aeqvales et ex eodem eivs latere, et qvadrilatervm verticvm $A$ , $B$ , $T'$ et $T$ rectangvlvm est. Vnaqvaeqve portio recta tantvm metitvr qvantvm latvs triangvli.
$\frac{\pi}{2} + \alpha + \frac{\pi}{2} + \varphi_A = 2\pi \rightarrow \dato{f}{\varphi_A = \pi - \alpha}$
In tvbo centri $A$ taenia arcvm amplectitvr inter pvncta contactvs dvarvm portionvm rectarvm qvae ad evm pervenivnt comprehensvm. Radii ad illa pvncta dvcti angvlos rectos cvm lateribvs $AB$ et $AC$ efficivnt; cvm angvlo $\alpha$ triangvli ad $A$ et angvlo ad centrvm $\varphi_A$ arcvs integram conversionem circa $A$ complent, ex svmma angvlorvm adiacentivm.
$r(\pi - \alpha) + r(\pi - \beta) + r(\pi - \gamma) = r\left( 3\pi - \resaltar{(\alpha + \beta + \gamma)} \right) = r\left( 3\pi - \resaltar{\pi} \right) = \dato{a}{2\pi r}$
Idem in reliqvis dvobvs tvbis accidit, cvm angvlis $\beta$ et $\gamma$ triangvli. Ex theoremate longitvdinis arcvs tres arcvs in svmmam efficivnt qvod infra scriptvm est, et cvm angvli triangvli in svmmam $\pi$ efficiant, simvl integram circvmferentiam efficivnt; hic vnvsqvisqve est tertia eivs portio, qvia $\alpha = \beta = \gamma = \frac{\pi}{3}$ .
$L = \resaltar{6r} + \resaltar{2\pi r}, \qquad r = 5 \rightarrow L = 30 + 10\pi$
Taenia metitvr svmmam portionvm rectarvm et arcvvm. Compvtatio arcvvm non adhibet triangvlvm aeqvilatervm esse: portio cvrva taeniae tensae circa tres tvbos aeqvales semper in svmmam integram circvmferentiam efficit. Pro $r = 5$ taenia circiter 61,4 cm metitvr.

Problema propositvm 4Rota qvae per saltvs progreditvr

Rota dentata in vnoqvoqve saltv per angvlvm $150^\circ$ contra motvm horologii convertitvr. Demonstrare eam primvm ad positionem initialem redire post $12$ saltvs, et in his saltibvs $12$ positiones distinctas occvpare. Generalivs: si vnvsqvisqve saltvs est fractio $\frac{p}{q}$ conversionis, cvm $p$ et $q$ inter se primis, qvot positiones distinctas occvpat?

Indicivm

Ex lemmate de angvlis coterminalibvs saltvs $j$ et $k$ rotam in eadem positione relinqvvnt si et solvm si $150(k - j)$ est mvltiplvm ipsivs $360$ , id est si $12$ dividit $5(k - j)$ . Adhibe $5$ et $12$ inter se primos esse, vt in lectione cvrsvs I de maximo commvni divisore et minimo commvni mvltiplo.

Responsvm

Dvo saltvs eandem positionem dant si et solvm si differentia eorvm est mvltiplvm ipsivs $12$ ; generatim rota $q$ positiones distinctas occvpat, et ad initialem primvm redit in saltv $q$ .

Solvtio explicata

Solvtio

$150(k - j) = 360m \leftrightarrow \dato{c}{5(k - j) = 12m}, \quad m \in \mathbb{Z}$
Post $k$ saltvs rota per $150k$ gradvs conversa est. Ex lemmate de angvlis coterminalibvs saltvs $j$ et $k$ eam in eadem positione relinqvvnt si et solvm si differentia angvlorvm est mvltiplvm integrvm vnivs conversionis; per $30$ dividendo, condicio est vt $5(k - j)$ sit mvltiplvm ipsivs $12$ .
$\resaltar{60 \mid 5(k - j)} \leftrightarrow \dato{d}{\resaltar{12 \mid k - j}}$
Nvmervs integer $5(k - j)$ tvm est mvltiplvm ipsivs $12$ et ipsivs $5$ , qvi inter se primi svnt; ex lemmate de mvltiplis commvnibvs lectionis cvrsvs I de maximo commvni divisore est mvltiplvm ipsivs $60$ , et $k - j$ est mvltiplvm ipsivs $12$ . Vice versa, si $k - j = 12t$ , angvli $5t$ conversionibvs differvnt.
$0 \leq j < k \leq 11 \rightarrow 0 < k - j < 12, \qquad 150 \cdot \resaltar{12} = \resaltar{5 \cdot 360}$
Inter saltvs $0$ et $11$ nvlla differentia est mvltiplvm ipsivs $12$ : rota $12$ positiones distinctas occvpat. In saltv $12$ per $1800^\circ$ conversa est, qvinqve integras conversiones, et primvm ad positionem initialem redit; inde easdem positiones eodem ordine iterat.
$q \mid p(k - j) \leftrightarrow \resaltar{q \mid k - j}$
Generatim, si vnvsqvisqve saltvs est fractio $\frac{p}{q}$ conversionis, saltvs $j$ et $k$ eandem positionem dant si et solvm si $\frac{p(k - j)}{q}$ integer est, id est si $q$ dividit $p(k - j)$ . Cvm $p$ et $q$ inter se primis, idem lemma dat $pq$ dividere $p(k - j)$ , et $q$ dividere $k - j$ : rota prorsvs $q$ positiones occvpat et ad initialem primvm redit in saltv $q$ . Fractio $\frac{150}{360} = \frac{5}{12}$ casvm svpra tractatvm dat.

Problema propositvm 5Indices ad angvlvm rectvm

Demonstrare, inter horam 12:00 et horam 12:00 proximam (hac excepta), indices horologii angvlvm rectvm prorsvs $22$ vicibvs efficere, et illa momenta invenire.

Indicivm

Notationibvs problematis solvti 5 adhibitis, indices angvlvm rectvm efficivnt cvm differentia $330t$ coterminalis est cvm $90^\circ$ vel cvm $270^\circ$ , id est cvm $330t = 90 + 180j$ pro aliqvo nvmero integro $j$ .

Responsvm

Momenta svnt qvae infra scripta svnt, in horis post horam 12:00, cvm $j = 0, 1, \ldots, 21$ ; primvm est circa horam 12:16 et inter ea svnt horae 3:00 et 9:00.

t = \frac{3(2j + 1)}{11}

Solvtio explicata

Solvtio

$\alpha_m - \alpha_h = 360t - 30t = \dato{d}{330t}$
Vt in problemate solvto 5, post $t$ horas index minvtorvm per $360t$ gradvs conversvs est et index horarvm per $30t$ , secvndvm motvm horologii ab hora 12:00; differentia positionvm eorvm est $330t$ gradvs.
$330t = 90 + 360m \vee 330t = 270 + 360m \leftrightarrow \dato{e}{330t = 90 + 180j}, \quad j \in \mathbb{Z}$
Indices angvlvm rectvm efficivnt si et solvm si illa differentia coterminalis est cvm $90^\circ$ vel cvm $270^\circ$ , id est si est $90 + 360m$ vel $270 + 360m$ pro aliqvo nvmero integro $m$ . Dvae familiae simvl svnt nvmeri $90 + 180j$ : valores pares ipsivs $j$ primam dant, impares alteram.
$t = \frac{\resaltar{90 + 180j}}{\resaltar{330}} = \dato{t}{\frac{3(2j + 1)}{11}}$
Solvendo pro $t$ et fractionem per $30$ simplificando momenta obtinentvr.
$0 \leq \frac{3(2j + 1)}{11} < 12 \leftrightarrow \resaltar{0 \leq j \leq 21}, \qquad j = 5 \rightarrow t = 3, \quad j = 16 \rightarrow t = 9$
Condicio $0 \leq t < 12$ aeqvivalet ipsi $0 \leq 2j + 1 < 44$ , id est ipsi $0 \leq j \leq 21$ : svnt prorsvs $22$ momenta, distincta qvia $t$ cvm $j$ crescit. Cvm $j = 0$ obtinetvr $t = \frac{3}{11}$ , circiter 16 minvta et 22 secvnda; cvm $j = 5$ et $j = 16$ , horae 3:00 et 9:00. Positiones respondentes indicis horarvm, $\frac{90(2j + 1)}{11}$ gradvs, in orbe horologii aeqvaliter distribvtae svnt.

Hac lectione angvli mensi et in plano collocati svnt. In proxima lectione, de rationibvs trigonometricis, coordinatae pvncti $P(\theta)$ circvmferentiae vnitariae cosinvm et sinvm cvivslibet angvli definient, et redvctio ad primam conversionem et angvli coterminales periodicitatem eorvm explicabvnt.

Angvli eorvmqve mensvra: gradvs et radiani

Svmmarivm

Proposita discendi

Angvli orientati

Systema sexagesimale

Radianvs

Conversio inter gradvs et radianos

Longitvdo arcvs et area sectoris

Angvlvs inscriptvs

Positio normalis et angvli coterminales

Velocitas angvlaris et velocitas linearis

Problemata solvta

Problemata proposita

Dvbitatio tva de hac lectione

Qvaestiones de hac lectione