I know that during the Renaissance, there was a romantisication of the old Romans and the Greeks, which led to a sort of revitalisation of the noble Arts and the sciences.
In the arts, they'd paint old Hellenic heroes or legends or Gods etc... and in the sciences the focus went back to a sort of humanism and philosophising (and thinking outside of Church doctrine) and researching the natural world again.
But what came first? Was there a sort of common reason both the arts and the sciences went through the Romance period, or did a couple of artists just focus on the Romans out of shared interest, and did this spread to others artists, like make the Romans popular amongst the "cultured nobility", which then influenced mainstream culture as well?
Like a small artistic minority that got "mainstream" and impacted society overal, like we can see with (bad but modern example) NWA etc?
The answer to you question, in a general sense, is that art and science in the Renaissance are two iterations of a common concern for the world as we perceive it through our senses. Be aware, in any case, that in the Renaissance you would not find the the of specialization in the workforce you see today: an artist and an engineer were often the same person. Also, the humanism and classical philosophy you ascribe to the sciences was also central to art production. Some goes for the study of the natural world—that's why you have artists like Dürer or Leonardo carefully sketching fauna and flora.