Yes, you're suppose to swing on the sword. I've been through it with no problem and other times couldn't swing worth jack. You may have to just rock back and forth until you are close enough to throw yourself to the other side. Sometimes I'd shoot a psi-blast at the sword-spitter right before jumping onto an already-stuck sword. Little Oly's stupid self dying isn't much help, either.
<No way you can read this. My handwriting's illegible>