Can I scale a div’s height proportionally to its width using CSS?

You can do it with the help of padding on a parent item, because relative padding (even height-wise) is based on the width of the parent element.

CSS:

.imageContainer {
    position: relative;
    width: 25%;
    padding-bottom: 25%;
    float: left;
    height: 0;
}

img {
    width: 100%;
    height: 100%;
    position: absolute;
    left: 0;
}

This is based on this article:
Proportional scaling of responsive boxes using just CSS