421211r
[Commentary:
This page is part of Harriot's account of the
refractive properties of the triangular prism.
Here, Harriot sets out the basic relations that hold among the angles of incidence and refraction, simply by reasons of geometry (i.e., independently of how one calculates the amount of refraction). He will use these identities throughout his treatment both of the simple refraction of light through the prism, and his
experiments on refractive colours, also conducted using triangular media. ]
Lemma. Propositio.
[Translation: Lemma. Proposition ]
[Translation: Lemma. Proposition ]
Duae perpendiculares super
duae lineas inclinantes
faciunt angulum aequalem
angulo inclinantium
[Translation: Two perpendicular lines constructed upon two intersecting lines make an angle to each other equal to the angle between the intersecting lines.
duae lineas inclinantes
faciunt angulum aequalem
angulo inclinantium
[Translation: Two perpendicular lines constructed upon two intersecting lines make an angle to each other equal to the angle between the intersecting lines.
Per angulum/prisma vitreum
[Translation: Through a glass angle/prism]
[Translation: Through a glass angle/prism]
Adfractivus et Abfractivus aerei sunt eiusdem amplitudinis
[???] Adfractus et Abfractivus vitrei sunt etiam aequales amplitudine.
[Translation: The adfractive and abfractive rays in air are of the same angular size.
The adfracted and abfractive rays in glass are also equal in angular size. ]
[Commentary: Here Harriot uses a variation of his usual terminology for the multiple angles of refraction through an interface and back out the other side. [Diagram here??]
: adfractivus (or incidens); translated 'adfractive'.
: adfractus; translated 'adfracted'.
: [incidens] abfractivus; translated 'abfractive'.
: abfractus; translated 'abfracted'.
Harriot states here, then, that , and — which is not true in general, but he will explain further below the circumstance in which this will be true. ]
[???] Adfractus et Abfractivus vitrei sunt etiam aequales amplitudine.
[Translation: The adfractive and abfractive rays in air are of the same angular size.
The adfracted and abfractive rays in glass are also equal in angular size. ]
[Commentary: Here Harriot uses a variation of his usual terminology for the multiple angles of refraction through an interface and back out the other side. [Diagram here??]
: adfractivus (or incidens); translated 'adfractive'.
: adfractus; translated 'adfracted'.
: [incidens] abfractivus; translated 'abfractive'.
: abfractus; translated 'abfracted'.
Harriot states here, then, that , and — which is not true in general, but he will explain further below the circumstance in which this will be true. ]
Dato igitur triangulo angulo (prismatis);
dabitur talis incidens adfractivus aereus
qui faciet abfractum aereum sibi amplitudine æqualem.
[Translation: And so, given the triangle angle (of the prism), just such an incident, adfractive ray in air will also be given, which will make the abfracted ray in air equal to it in angular size. ]
qui faciet abfractum aereum sibi amplitudine æqualem.
[Translation: And so, given the triangle angle (of the prism), just such an incident, adfractive ray in air will also be given, which will make the abfracted ray in air equal to it in angular size. ]
De medio densiori
[Translation: Concerning a denser interposed medium. ]
[Translation: Concerning a denser interposed medium. ]
Illud medium est superficierum parallelarum;
vel
[Translation: This medium has either parallel surfaces, or inclined ones. ]
vel
[Translation: This medium has either parallel surfaces, or inclined ones. ]
In medio planorum parallelorum, angulus adfractus
et abfractivus sunt
[Translation: In a medium [bounded by] plane parallel sides, the adfracted and abfractive angles are equal. ]
[Commentary: That is, the two angles within the glass are equal, or .
et abfractivus sunt
[Translation: In a medium [bounded by] plane parallel sides, the adfracted and abfractive angles are equal. ]
[Commentary: That is, the two angles within the glass are equal, or .
In medio planorum inclinantium, differentia vel summa angulorum
inclinationis, et adfracti, est amplitudo
[Translation: In a medium [bounded by] inclined plane sides, the difference or sum of the angle of inclination and the adfracted angle, is the angular size of the abfractive ray. ]
[Commentary: That is, if the sides are inclined at some angle (say ), then .
inclinationis, et adfracti, est amplitudo
[Translation: In a medium [bounded by] inclined plane sides, the difference or sum of the angle of inclination and the adfracted angle, is the angular size of the abfractive ray. ]
[Commentary: That is, if the sides are inclined at some angle (say ), then .
Angulus inclinationis
minor adfracto.
[Translation: Angle of inclination less than the adfracted [angle].
minor adfracto.
[Translation: Angle of inclination less than the adfracted [angle].
Summa. Si perpendiculares
secant se mutuo extra
angulum
[Translation: A sum: if the perpendiculars cut each other beyond the angle of inclination. ]
vide exempla magis
[Translation: See the more salient examples. ]
[Commentary: These more salient examples are found on f. 216.
secant se mutuo extra
angulum
[Translation: A sum: if the perpendiculars cut each other beyond the angle of inclination. ]
vide exempla magis
[Translation: See the more salient examples. ]
[Commentary: These more salient examples are found on f. 216.
maior adfracto.
[Translation: greater than the adfracted [angle]. ]
[Translation: greater than the adfracted [angle]. ]
Differentia. Se perpendiculares
secant se intra
angulum
[Translation: A difference: if the perpendiculars cut each other within the angle of inclination. ]
secant se intra
angulum
[Translation: A difference: if the perpendiculars cut each other within the angle of inclination. ]
aequalis
[Translation: equal to the adfracted [angle].
[Translation: equal to the adfracted [angle].
3 casus. secantes se in una
linearum continentium angulum inclinationis.
[Translation: Third case. Cutting each other in a single one of the line containing the angle of inclination. ]
linearum continentium angulum inclinationis.
[Translation: Third case. Cutting each other in a single one of the line containing the angle of inclination. ]
nota:
In medio rariori iidem canones
mutatis solum ad in ab &
[Translation: Note: In a rarer medium, the same rules [apply], just changing 'ad' to 'ab' and vice versa. ]
[Commentary: That is, swap 'abfractus' and 'adfractus', and 'abfractivus' and 'adfractivus', or and , and and ; so that, for example, the rule for a medium with sides inclined at angle becomes .
In medio rariori iidem canones
mutatis solum ad in ab &
[Translation: Note: In a rarer medium, the same rules [apply], just changing 'ad' to 'ab' and vice versa. ]
[Commentary: That is, swap 'abfractus' and 'adfractus', and 'abfractivus' and 'adfractivus', or and , and and ; so that, for example, the rule for a medium with sides inclined at angle becomes .