skip to content

Astronomical Images : Proportional parts of the superior planets

Peter Apian

Astronomical Images

<p style='text-align: justify;'>This Venetian edition of Peuerbach's <i>Theoricae novae </i>was copied from Apian's 1528 edition, printed in Ingolstadt. Subsequently, the work went through several further editions. Apian's edition added new woodcuts as well as notations to some of those from earlier editions. Some errors in the woodcuts in the 1528 edition were repeated in this Venetian edition of 1537. This small woodcut is a crude and reduced copy of a figure already present in the first edition of Peuerbach's treatise (c. 1474). It shows the proportional parts for the three superior planets. As in the case of the Moon, the proportional parts concern the calculation of the equation of the argument of the planet (<i>aequatio argumenti planetae</i>), defined as the arc of the zodiac lying between the true motion of the planet and the true motion of the epicycle. This equation of the argument is dependent on the true argument of the planet (<i>argumentum verum planetae</i>), measured on the circumference of the epicycle from the true apogee of the epicycle to the centre of the body of the planet. As the distance of the centre of the epicycle from the centre of the World varies, the diameter of this epicycle, measured on the zodiac from the centre of the World, also varies. Therefore, to the same value of the true argument of the planet (measured on the epicycle) correspond unequal arcs of equations (measured on the zodiac): 'greater when the centre of the epicycle is at the perigee of the deferent than when it is at the mean distances of the deferent. In the latter case they are again greater than when the centre of the epicycle is at the apogee of the deferent'. Still, as in the case of the Moon, the variations of the equations for the same given argument are called 'the variations of the diameter [of the small circle or epicycle]' (<i>diversitates diametri</i>). In the case of the superior planets, there are two kinds of 'variations of the diameter': the 'remoter or outer variations' (<i>diversitates diametri longiores</i>) are the excesses of the equations when the centre of the epicycle is at the mean longitudes of the deferent over the equations when it is at the apogee; the 'closer or inner variations' (<i>diversitates diametri propiores</i>) are the excesses of the equations when the centre of the epicycle is at the perigee over those when it is at the mean longitudes of the deferent. The 'remoter or outer proportional parts' (<i>minuta proportionalia longiora</i>) are the excesses of the line taken from the centre of the World to the apogee of the deferent over the line extended from the same centre to the mean longitude, divided into sixty equal parts. The 'closer or inner proportional parts' (<i>minuta proportionalia propiora</i>) are the excesses of the line taken from the centre of the World to the mean longitude over the line extended from the same centre to the perigee, divided into sixty equal parts. The diagram, though crude, small, unlabelled and regardless of proportions, shows the difference between these two kinds of proportional minutes. The outermost circle is drawn around the centre of the World; its radius extends from this centre to the apogee of the eccentric circle that bears the centre of the epicycle (the deferent circle). The innermost circle is also drawn around the centre of the World, and its radius extends from this centre to the perigee of the deferent circle. The vertical line is the line of the apogee. Two radii are drawn from the centre of the World to the points that determine the mean longitudes of the deferent. They separate the upper zone of the <i>minuta proportionalia longiora</i> from the lower zone of the <i>minuta proportionalia propiora</i>, each divided by concentric circles indicating the division into sixty minutes. The circle that marks the exterior limit of the blackened zones is the eccentric deferent of the epicycle. These blackened zones show that at the apogee the eccentric deferent circle encompasses all the 'remoter or outer proportional minutes', that it encompasses none of the 'closer or inner proportional minutes' at the perigee, and that in other places it encompasses some of them (either 'inner' or 'outer'): fewer near to the perigee, and more, in proportion, near to the apogee. When the centre of the epicycle is at the apogee, it is at sixty minutes, and at zero minutes at the perigee; in other places, it can be at ten, fifteen, or twenty proportional minutes, and so on. In Peuerbach's words, 'the equations of the arguments, which are written in the tables, occur when the centre of the epicycle is in the mean longitude, or distance, of the deferent' (<i>aequationes autem argumentorum, quae scribuntur in tabulis, contingunt centro epicycli in longitudine deferentis media constituto</i>). The tables do not give the values of the equation when the centre of the epicycle is in other places, but it is possible to reckon them. If we know the true centre (<i>centrum verum</i>) of the planet, the tables indicate the proportional parts; and if we know the true argument (<i>argumentum verum</i>) of the planet, we can calculate the value of the <i>diversitas diametri</i> (either <i>longior</i> or <i>propior</i>) corresponding to the exact position of the centre of the epicycle. 'The proportional part of this variation â?¦ is added to the equation of argument found in the tables, or subtracted from it' (<i>cuius diversitatis pars proportionalis â?¦ cum aequatione argumenti in tabula reperta addenda est, vel ab ea minuenda</i>). It is added when the planet is at or near the perigee (where the equations of the argument are greater than those at the mean longitudes), that is 'if the variation is closer or inner' (<i>si diversitas propior fuerit</i>), and it is subtracted near the apogee (where the equations of the argument are less than those at the mean longitudes), that is 'if it is remoter or outer' (<i>si longior</i>). Then we obtain 'the equation of the argument, both true and corrected' (<i>aequatio argumenti vera et aequata</i>), corresponding to the exact position of the centre of the epicycle. For an ameliorated version of this diagram, see Reinhold's 1553 commentary, fol. 57v. Translated quotations of Peuerbach's <i>Theoricae</i> are from Aiton (1987).</p>


Want to know more?

Under the 'More' menu you can find , and information about sharing this image.

No Contents List Available
No Metadata Available

Share

If you want to share this page with others you can send them a link to this individual page:
Alternatively please share this page on social media

You can also embed the viewer into your own website or blog using the code below: