Do I need permission to use an image from a book from 1942? The book is On growth and forms by D'arcy Thompson. It was published in 1942 and Thompson died in 1948. Wikipedia uses the image citing public domain. Can I use extracts and images from the book with proper citation without problem?

The image I want is on this page: 
https://archive.org/details/ongrowthform1917thom/page/322

As you can see the book is in public domain and is freely available on archive.org
It depends on whether or not the book is in the public domain in your specific country.  In some countries, copyright extends to some amount of time after the death of the author, for example 50 years.  However, your country's copyright laws may be different.