quae erant demonstranda

From Latin quae erant dēmonstranda (“which were to be proved”).

quae erant demonstranda

1. plural of quod erat demonstrandum
   • 1894, Benjamin Franklin Finkel, The American mathematical monthly: devoted to the interests of collegiate mathematics, volume 1, Mathematical Association of America, page 190:

     Therefore it is established that the join CD will be equal, or less, or greater than this AB, according as the angles at the same CD are right, or obtuse, or acute. Quae erant demonstranda.

   • 1955, Maurice Leonard Jacks, The education of good men, Gollancz, page 47:

     The proving of a geometrical theorem is, of course, an exercise in logical thinking, but the pupil needs to be made conscious of this, and also conscious of the fact that the same processes of thought are applicable in quite other fields and will lead to equally satisfactory results (quae erant demonstranda) there too.

quae (nominative neuter plural of quī) + erant (third-person plural imperfect active indicative of sum) + dēmōnstranda (nominative neuter plural of dēmōnstrandus, future participle of dēmōnstrō)

• (Classical Latin) IPA^(key): /kʷae̯ ˈe.rant deː.monˈstran.da/, [kʷäe̯ ˈɛrän̪t̪ d̪eːmõːˈs̠t̪rän̪d̪ä]
• (modern Italianate Ecclesiastical) IPA^(key): /kwe ˈe.rant de.monˈstran.da/, [kwɛː ˈɛːrän̪t̪ d̪emonˈst̪rän̪d̪ä]

quae erant dēmōnstranda

1. plural of quod erat dēmōnstrandum
   • 1986: Girolamo Saccheri, George Bruce Halsted (translator), Girolamo Saccheri’s Euclides vindicatus, page 24 (AMS Bookstore; ISBN 0‒8284‒0289‒2, 978‒0‒8284‒0289‒7)

     Itaque constat junctam CD aequalem fore, aut mino- ^[4] rem, aut majorem ipsa AB, prout anguli ad eandem CD fuerint aut recti, aut obtusi, aut acuti. Quae erant demonstranda.

     (please add an English translation of this quotation)